#7502208

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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2225 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)(13,14,17,21)(15,23,25,19)(16,20,28,36)(18,22,30,38)(24,41,42,33)(26,39,31,34)(27,44,43,35)(29,37,32,40);(6,9)(7,12)(13,15,17,25)(14,19,21,23)(16,27,28,43)(18,31,30,26)(20,35,36,44)(22,39,38,34)(24,32,42,29)(33,40,41,37);(2,4)(13,16,17,28)(14,20,21,36)(15,24,25,42)(18,32,30,29)(19,33,23,41)(22,40,38,37)(26,43,31,27)(34,44,39,35);(2,4)(5,7,8,12)(6,10,9,11)(14,21)(15,25)(20,36)(22,38)(24,42)(26,31)(27,43)(37,40);(14,21)(19,23)(20,36)(22,38)(33,41)(34,39)(35,44)(37,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,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, 7409404);