#6519941

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
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	#28584	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6427137);