#6463111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2499 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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, 6370307);