#6434030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1741 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(19,26)(20,21)(23,31)(24,30)(27,32)(28,29);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,27)(20,30)(21,24)(22,25)(23,29)(26,32)(28,31);(11,15)(13,16)(17,19,21,26,22,29,20,28)(18,23,24,31,25,27,30,32);(10,14)(13,16)(17,20,22,21)(18,24,25,30)(19,28,29,26)(23,31,27,32);(1,3,4,7)(2,6,5,8)(19,29)(23,27)(26,28)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257274, #72257462, #72257627, …
maximal normal subgroup of	#72257274, #72257462, #72257627, …
maximal quotient of	#72265338, #72320369

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341226);