#7562938

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#45159	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1807 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=gi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470134);