#6467540

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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

2249 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)(13,14,17,23,18,24,19,20)(15,25,29,34,28,21,26,37)(16,22,32,40,33,38,30,35)(27,39,42,43,31,36,41,44);(2,4)(14,20)(15,26)(16,30)(17,19)(21,25)(22,38)(23,24)(27,31)(28,29)(32,33)(36,44)(39,43);(6,9)(10,12)(13,15,19,26,18,28,17,29)(14,21,20,34,24,25,23,37)(16,31,30,42,33,27,32,41)(22,39,35,44,38,36,40,43);(13,16,19,30,18,33,17,32)(14,22,20,35,24,38,23,40)(15,27,26,41,28,31,29,42)(21,36,34,43,25,39,37,44);(5,7,8,11)(6,10,9,12)(15,28)(21,25)(26,29)(27,31)(34,37)(36,39)(41,42)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374736);