#7592922

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 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)(13,14)(15,17)(16,19)(18,26)(20,24)(21,25)(22,28)(23,27);(2,3)(5,7)(6,9)(8,11)(10,12)(13,15,18,24,19,23,28,21)(14,20,22,17,16,25,26,27);(1,2,4,3)(13,16,19,14)(15,20,23,25)(17,21,27,24)(18,26,28,22);(1,3,4,2)(7,11)(9,12)(14,16)(15,23)(21,24)(22,26);(13,17,18,25,19,27,28,20)(14,21,22,15,16,24,26,23)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=g,d^2=gi,e^2=f,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

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, 7500118);