#7519982

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1861 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)(8,12)(9,10)(13,14,20,26)(15,21,31,39)(16,22,17,23)(18,24,19,25)(27,35,33,41)(28,36,43,44)(29,37,30,38)(32,40,34,42);(1,2)(3,4)(6,10)(11,12)(13,15,20,31)(14,21,26,39)(16,27,17,33)(18,30,19,29)(22,35,23,41)(24,38,25,37)(28,32,43,34)(36,40,44,42);(2,4)(5,7,9,8)(6,11,10,12)(13,16,20,17)(14,22,26,23)(15,27,31,33)(18,34,19,32)(21,35,39,41)(24,42,25,40)(28,29,43,30)(36,37,44,38);(2,4)(13,17,20,16)(14,23,26,22)(15,28,31,43)(18,34,19,32)(21,36,39,44)(24,42,25,40)(27,29,33,30)(35,37,41,38);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,24,26,25)(15,29,31,30)(16,32,17,34)(21,37,39,38)(22,40,23,42)(27,28,33,43)(35,36,41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,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, 7427178);