#7561621

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
prime radical	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2163 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7468817);