#6467483

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	#25777	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2316 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)(13,14,18,25,19,24,20,26)(15,27,30,39,31,40,32,21)(16,33,17,23,34,22,35,42)(28,41,44,37,36,38,29,43);(2,4)(13,15)(14,21)(16,29)(17,36)(18,32)(19,31)(20,30)(22,41)(23,37)(24,39)(25,40)(26,27)(28,35)(33,38)(34,44)(42,43);(6,9)(10,12)(13,16,20,35,19,34,18,17)(14,22,26,23,24,33,25,42)(15,28,32,29,31,36,30,44)(21,37,40,41,39,43,27,38);(13,17,18,34,19,35,20,16)(14,23,25,22,24,42,26,33)(15,29,30,28,31,44,32,36)(21,38,27,43,39,41,40,37);(5,7,8,11)(6,10,9,12)(14,24)(21,39)(22,33)(23,42)(25,26)(27,40)(37,43)(38,41)

Polycyclic presentation:

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

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, 6374679);