#6434031

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
prime radical	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1619 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)(17,18,22,26)(19,27,31,24)(20,21,25,23)(28,30,32,29);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,32)(21,30)(22,31)(23,29)(25,28)(26,27);(11,15)(13,16)(17,20,23,18,22,25,21,26)(19,28,29,27,31,32,30,24);(10,14)(13,16)(17,21,22,23)(18,20,26,25)(19,29,31,30)(24,28,27,32);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,25,26,20)(19,30,31,29)(24,28,27,32)

Polycyclic presentation:

2^9|a^2=i,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	#72257583, #72264121, #72264267, …
maximal normal subgroup of	#72257583, #72264121, #72264267, …
maximal quotient of	#72265097

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341227);