#7517041

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1990 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)(15,20,23,18)(16,25,30,37)(17,26,34,39)(21,31,27,29)(22,33,40,35)(28,36,32,38)(41,43,42,44);(1,2)(3,4)(6,9)(10,12)(13,15,19,23)(14,18,24,20)(16,27,30,21)(17,22,34,40)(25,31,37,29)(26,35,39,33)(28,42,32,41)(36,43,38,44);(13,16,19,30)(14,21,24,27)(15,25,23,37)(17,32,34,28)(18,31,20,29)(22,38,40,36)(26,42,39,41)(33,43,35,44);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,34)(14,22,24,40)(15,26,23,39)(16,28,30,32)(18,33,20,35)(21,36,27,38)(25,41,37,42)(29,43,31,44);(6,9)(10,12)(13,18,19,20)(14,23,24,15)(16,29,30,31)(17,33,34,35)(21,25,27,37)(22,39,40,26)(28,43,32,44)(36,41,38,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^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, 7424237);