#7514666

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2715 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)(7,11)(8,10)(9,12)(13,14)(15,23)(16,27)(17,19)(18,32)(20,24)(21,28)(22,31)(25,38)(26,37)(29,40)(30,39)(33,35)(34,36)(41,43)(42,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(15,24)(16,28)(18,31)(20,37)(21,39)(23,35)(26,33)(27,36)(30,34)(38,40)(42,43);(13,15,17,26)(14,20,19,35)(16,29,30,25)(18,33,31,24)(21,40,36,38)(22,37,32,23)(27,44,39,41)(28,43,34,42);(2,4)(13,16,17,30)(14,21,19,36)(15,25,26,29)(18,34,31,28)(20,38,35,40)(22,39,32,27)(23,41,37,44)(24,42,33,43);(6,10)(9,11)(14,19)(16,30)(20,35)(22,32)(23,37)(25,29)(28,34)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421862);