#7502407

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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

2223 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)(14,19)(16,27)(20,33)(22,31)(23,36)(24,38)(29,41)(40,44);(6,9)(7,12)(13,14,17,19)(15,23,25,36)(16,28,27,21)(18,31,30,22)(20,32,33,26)(24,39,38,37)(29,35,41,43)(34,40,42,44);(2,4)(13,15,17,25)(14,20,19,33)(16,24,27,38)(18,32,30,26)(21,34,28,42)(22,36,31,23)(29,44,41,40)(35,39,43,37);(2,4)(5,7,8,12)(6,10,9,11)(13,16,17,27)(14,21,19,28)(15,24,25,38)(18,29,30,41)(20,34,33,42)(22,35,31,43)(23,37,36,39)(26,40,32,44);(16,27)(21,28)(24,38)(29,41)(34,42)(35,43)(37,39)(40,44)

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,c^b=ch,d^b=df,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, 7409603);