#6454610

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3031 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361806);