#6434235

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341431);