#6467549

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2243 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)(13,14,19,20,18,24,17,23)(15,25,33,38,28,21,26,35)(16,29,34,40,32,22,30,36)(27,37,42,43,31,39,41,44);(2,4)(14,20)(15,26)(16,30)(17,19)(23,24)(27,31)(28,33)(32,34)(35,38)(36,40)(37,44)(39,43);(6,9)(10,12)(13,15,17,26,18,28,19,33)(14,21,23,38,24,25,20,35)(16,31,30,42,32,27,34,41)(22,39,40,43,29,37,36,44);(13,16,17,30,18,32,19,34)(14,22,23,40,24,29,20,36)(15,27,26,41,28,31,33,42)(21,37,38,44,25,39,35,43);(5,7,8,11)(6,10,9,12)(15,28)(21,25)(26,33)(27,31)(35,38)(37,39)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=h,e^2=gi,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, 6374745);