#6465811

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2001 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263899, #72263949
maximal normal subgroup of	#72263899, #72263949
maximal quotient of	#72264864

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373007);