#7593008

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1949 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=gi,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=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, 7500204);