#6483121

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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

2012 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)(13,14,19,24)(15,20,26,35)(16,28,32,38)(17,23,18,22)(21,30,39,31)(25,41,29,43)(27,40,33,36)(34,44,37,42);(2,4)(7,8)(10,12)(13,15)(14,20)(16,29)(17,33)(18,27)(19,26)(21,37)(22,40)(23,36)(24,35)(25,32)(28,43)(30,42)(31,44)(34,39)(38,41);(5,7,9,8)(6,10,11,12)(13,16,17,30,19,32,18,31)(14,21,22,38,24,39,23,28)(15,25,33,44,26,29,27,42)(20,34,40,43,35,37,36,41);(5,8,9,7)(6,10,11,12)(15,26)(20,35)(25,29)(27,33)(34,37)(36,40)(41,43)(42,44);(6,11)(10,12)(13,17,19,18)(14,22,24,23)(15,27,26,33)(16,30,32,31)(20,36,35,40)(21,38,39,28)(25,42,29,44)(34,41,37,43)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=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, 6390317);