#7611531

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

2058 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)(16,21)(17,19)(18,22)(23,32)(24,29)(25,33)(26,35)(27,31)(28,30)(34,39)(36,43)(37,41)(38,40)(42,44);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(14,19)(16,26)(18,27)(20,29)(21,34)(23,36)(25,37)(28,39)(30,35)(32,42)(38,44)(40,43);(6,10)(7,11)(13,15)(14,20)(16,23)(17,24)(18,25)(19,29)(21,32)(22,33)(26,36)(27,37)(28,38)(30,40)(31,41)(34,42)(35,43)(39,44);(6,10)(9,12)(13,16,27,39,17,28,18,26)(14,21,31,34,19,30,22,35)(15,23,37,44,24,38,25,36)(20,32,41,42,29,40,33,43);(1,3,4,2)(14,19)(15,24)(16,28)(22,31)(25,37)(26,39)(32,40)(42,43)

Polycyclic presentation:

2^9|b^2=fi,d^2=hi,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,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7518727);