#7522386

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

2178 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)(8,10)(9,12)(13,14,17,21)(15,23,26,19)(16,20,28,34)(18,22,30,36)(24,37,39,31)(25,33,27,32)(29,35,41,44)(38,43,40,42);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,26)(14,19,21,23)(16,24,28,39)(18,25,30,27)(20,31,34,37)(22,32,36,33)(29,38,41,40)(35,42,44,43);(2,4)(13,16)(14,20)(15,24)(17,28)(18,29)(19,31)(21,34)(22,35)(23,37)(25,38)(26,39)(27,40)(30,41)(32,42)(33,43)(36,44);(2,4)(5,7,9,8)(6,11,12,10)(15,25)(18,30)(19,32)(22,36)(23,33)(24,38)(26,27)(29,41)(31,42)(35,44)(37,43)(39,40);(5,8,9,7)(6,11,12,10)(14,21)(16,28)(19,23)(22,36)(24,39)(29,41)(32,33)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,e^2=fi,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, 7429582);