#6467545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2239 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)(13,14,18,20,19,24,17,23)(15,21,25,34,27,36,28,38)(16,29,30,40,33,22,32,35)(26,39,41,44,31,37,42,43);(2,4)(14,20)(15,25)(16,30)(17,18)(21,36)(22,29)(23,24)(26,31)(27,28)(32,33)(37,44)(39,43);(6,9)(10,12)(13,15,18,25,19,27,17,28)(14,21,20,34,24,36,23,38)(16,31,30,42,33,26,32,41)(22,39,35,44,29,37,40,43);(13,16,18,30,19,33,17,32)(14,22,20,35,24,29,23,40)(15,26,25,41,27,31,28,42)(21,37,34,43,36,39,38,44);(5,7,8,11)(6,10,9,12)(15,27)(21,36)(25,28)(26,31)(34,38)(37,39)(41,42)(43,44)

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,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei

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, 6374741);