#7522426

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

2226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fi,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, 7429622);