#6467501

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2263 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)(7,10)(8,9)(11,12)(13,14,20,35,19,24,18,25)(15,26,34,21,30,42,29,38)(16,22,33,41,32,40,17,23)(27,39,28,36,43,37,31,44);(2,4)(13,15)(14,21)(16,31)(17,27)(18,34)(19,30)(20,29)(22,39)(23,36)(24,38)(25,42)(26,35)(28,32)(33,43)(37,40)(41,44);(6,9)(10,12)(13,16,18,17,19,32,20,33)(14,22,25,23,24,40,35,41)(15,27,29,31,30,43,34,28)(21,36,26,39,38,44,42,37);(13,17,18,32,19,33,20,16)(14,23,25,40,24,41,35,22)(15,28,29,27,30,31,34,43)(21,37,26,36,38,39,42,44);(5,7,8,11)(6,10,9,12)(14,24)(21,38)(22,40)(23,41)(25,35)(26,42)(36,44)(37,39)

Polycyclic presentation:

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

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, 6374697);