#7564923

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	#59053	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1994 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=g,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,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, 7472119);