#7611579

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

2044 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)(15,20)(16,26)(17,19)(18,22)(21,30)(23,36)(24,31)(25,34)(27,35)(28,33)(29,32)(37,41)(38,43)(39,42)(40,44);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(14,19)(16,27)(18,29)(20,31)(21,35)(23,38)(25,39)(26,37)(28,40)(30,41)(33,43)(36,44);(6,10)(7,11)(13,15)(14,20)(16,28)(17,24)(18,25)(19,31)(21,36)(22,34)(23,30)(26,33)(27,40)(29,39)(32,42)(35,44)(37,43)(38,41);(6,10)(9,12)(13,16,29,41,17,30,18,27)(14,21,32,35,19,26,22,37)(15,23,39,40,24,28,25,38)(20,33,42,43,31,36,34,44);(1,3,4,2)(14,19)(15,24)(16,30)(22,32)(25,39)(27,41)(33,36)(43,44)

Polycyclic presentation:

2^9|b^2=fi,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7518775);