#6657468

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37305	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,5)(3,7,8,6)(11,15)(14,16)(17,18)(19,23)(20,28)(21,29)(22,25)(24,31)(26,30)(27,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,24)(18,23,25,31)(20,29,32,26)(21,27,30,28);(2,5)(6,7)(10,13)(14,16)(17,20,30,23,22,32,21,31)(18,24,29,27,25,19,26,28);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,30)(29,32);(9,11,12,15)(10,14,13,16)(18,25)(19,24)(26,29)(27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6564664);