#7518470

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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1981 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)(7,11)(8,9)(10,12)(13,14,18,23)(15,24,27,20)(16,28,31,21)(17,32,19,22)(25,35,43,41)(26,36,33,37)(29,38,42,44)(30,39,34,40);(1,2)(3,4)(6,9)(10,11)(13,15,18,27)(14,20,23,24)(16,29,31,42)(17,26,19,33)(21,38,28,44)(22,36,32,37)(25,34,43,30)(35,39,41,40);(2,4)(13,15)(14,20)(16,25)(17,33)(18,27)(19,26)(21,35)(22,37)(23,24)(28,41)(29,30)(31,43)(32,36)(34,42)(38,40)(39,44);(2,4)(13,16,18,31)(14,21,23,28)(15,25,27,43)(17,30,19,34)(20,35,24,41)(22,39,32,40)(26,42,33,29)(36,38,37,44);(5,7,8,12)(6,10,9,11)(13,17,18,19)(14,22,23,32)(15,26,27,33)(16,30,31,34)(20,36,24,37)(21,39,28,40)(25,42,43,29)(35,38,41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7425666);