#7560675

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	#36015	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1840 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)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,26,34)(16,20,27,37)(17,22,31,38)(23,32,41,44)(24,33,42,43)(25,35,29,39)(28,36,30,40);(2,3)(6,9)(10,12)(13,15,18,26)(14,19,21,34)(16,23,27,41)(17,29,31,25)(20,32,37,44)(22,39,38,35)(24,30,42,28)(33,40,43,36);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16,18,27)(14,20,21,37)(15,23,26,41)(17,28,31,30)(19,32,34,44)(22,36,38,40)(24,25,42,29)(33,35,43,39);(1,3,4,2)(13,16)(14,20)(15,24)(17,30)(18,27)(19,33)(21,37)(22,40)(23,29)(25,41)(26,42)(28,31)(32,39)(34,43)(35,44)(36,38);(6,9)(10,12)(14,21)(16,27)(19,34)(22,38)(23,41)(24,42)(28,30)(35,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467871);