#6517105

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500443
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	#37550	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2187 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)(13,14)(15,24)(16,28)(17,22)(18,34)(19,23)(20,27)(21,31)(25,42)(26,37)(29,41)(30,40)(32,36)(33,39)(35,43)(38,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,27,19,26,17,32)(14,20,22,24,23,36,34,37)(16,29,30,43,31,25,33,44)(21,38,40,42,28,35,39,41);(2,4)(6,9)(7,11)(13,16)(14,21)(15,25)(17,33)(18,30)(19,31)(20,35)(22,40)(23,28)(24,41)(26,29)(27,44)(32,43)(34,39)(36,38)(37,42);(5,7,8,11)(6,10,9,12)(15,26)(20,36)(24,37)(25,29)(27,32)(35,38)(41,42)(43,44);(6,9)(10,12)(13,17,19,18)(14,22,23,34)(15,27,26,32)(16,30,31,33)(20,37,36,24)(21,39,28,40)(25,43,29,44)(35,41,38,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6424301);