#6684961

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#39357	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2035 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592157);