#7560954

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	C₂×#3326	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,26,29,21)(16,30,28,38)(17,32,35,42)(18,25,20,23)(22,36,41,34)(27,39,43,44)(31,40,33,37);(1,2)(3,4)(6,9)(10,12)(13,15)(14,21)(16,20)(17,33)(18,28)(19,29)(22,39)(23,38)(24,26)(25,30)(27,36)(31,35)(32,40)(34,43)(37,42)(41,44);(5,7,8,11)(6,10,9,12)(13,16,20,15,19,28,18,29)(14,21,25,38,24,26,23,30)(17,33,36,43,35,31,34,27)(22,40,42,44,41,37,32,39);(2,4)(13,17,20,36,19,35,18,34)(14,22,25,42,24,41,23,32)(15,27,28,33,29,43,16,31)(21,37,38,39,26,40,30,44);(6,9)(10,12)(13,18,19,20)(14,23,24,25)(15,28,29,16)(17,34,35,36)(21,38,26,30)(22,32,41,42)(27,33,43,31)(37,39,40,44)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=gi,e^2=i,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, 7468150);