#7561763

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	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2211 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=fg,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, 7468959);