#6467563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#25797	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2286 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)(13,14,18,24,20,26,19,25)(15,27,30,39,32,38,31,21)(16,22,17,23,33,40,34,41)(28,43,44,42,35,37,29,36);(2,4)(13,15,20,32)(14,21,26,39)(16,29,33,44)(17,35,34,28)(18,31,19,30)(22,37,40,43)(23,42,41,36)(24,38,25,27);(6,9)(10,12)(13,16,18,17,20,33,19,34)(14,22,24,23,26,40,25,41)(15,28,30,44,32,35,31,29)(21,36,27,43,39,42,38,37);(13,17,18,33,20,34,19,16)(14,23,24,40,26,41,25,22)(15,29,30,28,32,44,31,35)(21,37,27,36,39,43,38,42);(5,7,8,11)(6,10,9,12)(16,33)(17,34)(22,40)(23,41)(28,35)(29,44)(36,42)(37,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,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,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374759);