#7502252

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,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, 7409448);