#246973

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 961 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,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, 154169);