#403395

order	512
nilpotency class	3
isoclinism	isoclinic class of #393893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=g,c^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,h^b=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268165, #72268167, #72268169
maximal normal subgroup of	#72268165, #72268167, #72268169

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310591);