#6462940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2286 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di

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, 6370136);