#7515402

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1947 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)(15,20,26,39)(16,28,30,44)(17,32,25,41)(18,34,37,23)(21,36,42,31)(22,29,38,33)(27,43,35,40);(1,2)(3,4)(6,9)(10,12)(13,15,19,26)(14,20,24,39)(16,29,30,33)(17,31,25,36)(18,35,37,27)(21,41,42,32)(22,28,38,44)(23,43,34,40);(5,7,8,11)(6,10,9,12)(13,16,19,30)(14,21,24,42)(15,25,26,17)(18,36,37,31)(20,38,39,22)(23,44,34,28)(27,33,35,29)(32,43,41,40);(2,4)(13,17,19,25)(14,22,24,38)(15,16,26,30)(18,29,37,33)(20,21,39,42)(23,41,34,32)(27,36,35,31)(28,40,44,43);(6,9)(10,12)(15,26)(16,30)(20,39)(21,42)(27,35)(28,44)(31,36)(40,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^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,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7422598);