#6451724

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3932 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)(7,12)(8,10)(9,11)(13,14,19,24)(15,25,29,40)(16,21,31,38)(17,22,32,39)(18,33,30,23)(20,34,37,28)(26,36,42,44)(27,41,43,35);(1,2)(3,4)(13,15)(14,20)(16,26)(17,27)(18,34)(19,29)(21,35)(22,36)(23,40)(24,37)(25,33)(28,30)(31,42)(32,43)(38,41)(39,44);(2,4)(5,7,9,8)(6,10,11,12)(13,16,19,31)(14,21,24,38)(15,26,29,42)(17,30,32,18)(20,35,37,41)(22,23,39,33)(25,36,40,44)(27,28,43,34);(2,4)(5,8,9,7)(6,10,11,12)(13,17)(14,22)(15,27)(16,30)(18,31)(19,32)(20,36)(21,23)(24,39)(25,41)(26,28)(29,43)(33,38)(34,42)(35,40)(37,44);(16,31)(17,32)(21,38)(22,39)(26,42)(27,43)(35,41)(36,44)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6358920);