#7519942

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

1859 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)(15,21,30,38)(16,22,17,23)(18,24,19,25)(27,35,31,39)(28,36,43,44)(29,37,34,42)(32,40,33,41);(1,2)(3,4)(6,10)(11,12)(13,15)(14,21)(16,31)(17,27)(18,29)(19,34)(20,30)(22,39)(23,35)(24,37)(25,42)(26,38)(28,33)(32,43)(36,41)(40,44);(2,4)(5,7,9,8)(6,11,10,12)(13,16,20,17)(14,22,26,23)(15,27,30,31)(18,33,19,32)(21,35,38,39)(24,41,25,40)(28,29,43,34)(36,37,44,42);(2,4)(13,17,20,16)(14,23,26,22)(15,28,30,43)(18,33,19,32)(21,36,38,44)(24,41,25,40)(27,29,31,34)(35,37,39,42);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,24,26,25)(15,29,30,34)(16,32,17,33)(21,37,38,42)(22,40,23,41)(27,28,31,43)(35,36,39,44)

Polycyclic presentation:

2^9|a^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=cgi,d^b=dh,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, 7427138);