#247038

order	512
nilpotency class	3
isoclinism	isoclinic class of #247017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#51754	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

993 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 993 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,16)(3,17,18,11)(4,23,26,40)(5,28,29,52)(6,10,36,45)(8,15,35,14)(9,27,41,20)(12,24,46,19)(13,37,44,33)(21,42,55,54)(22,48,30,49)(25,39,34,63)(31,62,61,51)(32,53,58,47)(38,60,64,57)(43,59,50,56);(1,3,8,22)(2,9,15,40)(4,24,27,29)(5,26,19,20)(6,32,37,55)(7,18,35,30)(10,39,44,64)(11,46,49,52)(12,48,28,17)(13,38,45,63)(14,23,16,41)(21,36,58,33)(25,31,60,56)(34,61,57,59)(42,51,47,50)(43,54,62,53);(2,10)(3,18)(5,29)(6,33)(9,38)(11,42)(12,50)(13,14)(15,44)(16,45)(17,54)(19,24)(21,58)(22,30)(23,39)(25,57)(28,51)(31,56)(32,55)(34,60)(36,37)(40,63)(41,64)(43,46)(47,49)(48,53)(52,62)(59,61);(1,4,7,26)(2,11,16,17)(3,19,18,24)(5,30,29,22)(6,34,36,25)(8,27,35,20)(9,12,41,46)(10,42,45,54)(13,53,44,47)(14,48,15,49)(21,59,55,56)(23,52,40,28)(31,58,61,32)(33,60,37,57)(38,50,64,43)(39,62,63,51)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154234);