#6467538

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	#25793	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2315 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,23,20,25,19,24)(15,26,28,37,29,36,17,21)(16,22,33,41,32,40,34,42)(27,43,31,39,44,38,30,35);(2,4)(13,15,20,29)(14,21,25,37)(16,30,32,31)(17,19,28,18)(22,38,40,43)(23,36,24,26)(27,33,44,34)(35,41,39,42);(6,9)(10,12)(13,16,19,34,20,32,18,33)(14,22,24,42,25,40,23,41)(15,27,17,30,29,44,28,31)(21,35,36,38,37,39,26,43);(13,17,19,29,20,28,18,15)(14,21,24,36,25,37,23,26)(16,31,34,27,32,30,33,44)(22,39,42,43,40,35,41,38);(5,7,8,11)(6,10,9,12)(16,32)(22,40)(27,44)(30,31)(33,34)(35,39)(38,43)(41,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=g,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, 6374734);