#7611922

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#45134	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2046 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)(7,9)(8,10)(11,12)(13,14,17,19)(15,20,24,29)(16,21,27,30)(18,22,28,31)(23,32,37,40)(25,33,38,41)(26,35,39,34)(36,43,44,42);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(14,19)(16,26)(18,28)(20,29)(21,34)(23,36)(25,38)(27,39)(30,35)(32,42)(37,44)(40,43);(6,10)(7,11)(13,15,17,24)(14,20,19,29)(16,23,27,37)(18,25,28,38)(21,32,30,40)(22,33,31,41)(26,36,39,44)(34,42,35,43);(6,10)(9,12)(13,16,18,26,17,27,28,39)(14,21,22,35,19,30,31,34)(15,23,25,36,24,37,38,44)(20,32,33,43,29,40,41,42);(1,3,4,2)(14,19)(15,24)(21,30)(22,31)(23,37)(25,38)(34,35)(36,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7519118);