#246986

order	512
nilpotency class	3
isoclinism	isoclinic class of #246957
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	#18156	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

986 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fi,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, 154182);