#6458774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35273	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1878 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)(9,10,12,14)(11,13,15,16)(17,18,24,27)(19,22,26,23)(20,29,21,25)(28,30,31,32);(9,11)(10,13)(12,15)(14,16)(17,19,23,18,24,26,22,27)(20,28,30,25,21,31,32,29);(2,5)(6,8)(11,15)(13,16)(17,20,24,21)(18,25,27,29)(19,28,26,31)(22,32,23,30);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21,24,20)(18,25,27,29)(19,28,26,31)(22,30,23,32);(17,22,24,23)(18,26,27,19)(20,30,21,32)(25,28,29,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365970);