#6444904

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2278 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)(13,14,17,19)(15,22,24,28)(16,21,18,26)(20,23,27,25);(1,2)(3,4)(6,9)(10,12)(14,19)(20,27)(21,26)(22,28);(2,4)(13,15,18,25,17,24,16,23)(14,20,21,22,19,27,26,28);(2,4)(13,16,17,18)(14,21,19,26)(15,23,24,25)(20,22,27,28);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343298, #72343301, #72343306, …
maximal normal subgroup of	#72343298, #72343301, #72343306, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352100);