#7502216

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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2194 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:

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

Polycyclic presentation:

2^9|a^2=fi,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

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, 7409412);