#6684893

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2014 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:

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592089);