#232904

order	512
nilpotency class	3
isoclinism	isoclinic class of #232897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1160 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)(3,12)(4,9)(5,16)(6,10)(7,11)(8,13)(14,15)(17,18)(19,29)(20,33)(21,28)(22,27)(23,26)(24,31)(25,36)(30,38)(32,43)(34,41)(35,42)(37,39)(40,45)(44,46)(47,48);(1,3)(2,8)(4,11)(5,9)(6,13)(7,14)(10,15)(12,16)(17,19)(18,24)(20,30)(21,37)(22,31)(23,32)(25,40)(26,39)(27,43)(28,29)(33,44)(34,45)(35,47)(36,46)(38,42)(41,48);(1,4)(2,9)(3,10)(5,15)(6,12)(7,8)(11,13)(14,16)(17,20)(18,25)(19,30)(21,36)(22,35)(23,34)(24,38)(26,42)(27,33)(28,41)(29,44)(31,47)(32,45)(37,46)(39,40)(43,48);(1,5,7,6)(2,10,11,16)(3,9,14,13)(4,12,8,15)(18,26)(20,34)(24,39)(27,28)(29,43)(30,45)(35,36)(46,47)

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=ghi,d^c=dh,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140100);