#232795

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,31)(25,37)(30,46)(32,41)(33,40)(35,45)(36,44)(38,43)(39,47)(42,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,15)(10,16)(17,19)(18,24)(20,35)(21,32)(22,31)(23,33)(25,42)(26,29)(27,40)(28,41)(30,36)(34,39)(37,47)(38,48)(43,46)(44,45);(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,39)(26,34)(27,43)(28,44)(29,45)(31,35)(32,48)(33,47)(40,42)(41,46);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,16,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,40,41,29)(25,34,44,43)(30,35,47,48)(39,42,46,45)

Polycyclic presentation:

2^9|d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139991);