#7564227

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
prime radical	#28916	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1832 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)(7,11)(8,12)(9,10)(13,14,17,19)(15,16,24,26)(18,27,23,22)(20,28,25,21);(1,2)(3,4)(6,10)(11,12)(14,19)(15,21)(16,25)(18,23)(20,26)(24,28);(5,7,9,8)(6,11,10,12)(13,15,23,28,17,24,18,21)(14,20,27,26,19,25,22,16);(2,4)(13,16,18,25,17,26,23,20)(14,21,22,24,19,28,27,15);(5,8,9,7)(6,11,10,12)(14,19)(15,24)(21,28)(22,27)

Polycyclic presentation:

2^9|a^2=i,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,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, 7471423);