#6685013

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2013 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:

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

Polycyclic presentation:

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

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, 6592209);