#6459506

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#35150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2427 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=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, 6366702);