#7516238

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
prime radical	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2033 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)(7,10)(8,9)(11,12)(13,14,17,21)(15,23,26,19)(16,20,28,36)(18,22,30,38)(24,41,43,33)(25,42,44,34)(27,39,31,35)(29,37,32,40);(1,2)(3,4)(6,9)(10,12)(13,15,17,26)(14,19,21,23)(16,24,28,43)(18,31,30,27)(20,33,36,41)(22,39,38,35)(25,32,44,29)(34,40,42,37);(2,4)(13,16)(14,20)(15,24)(17,28)(18,29)(19,33)(21,36)(22,37)(23,41)(25,31)(26,43)(27,44)(30,32)(34,39)(35,42)(38,40);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,25)(17,28)(18,32)(19,34)(21,36)(22,40)(23,42)(24,31)(26,44)(27,43)(29,30)(33,39)(35,41)(37,38);(6,9)(10,12)(16,28)(20,36)(24,43)(25,44)(29,32)(33,41)(34,42)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7423434);