#6672756

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3305	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2046 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)(13,14,19,17)(15,20,27,24)(16,21,30,31)(18,22,34,32)(23,35,41,42)(25,36,26,37)(28,38,43,44)(29,39,33,40);(6,9)(7,11)(13,15)(14,20)(16,28)(17,24)(18,25)(19,27)(21,38)(22,36)(23,29)(26,34)(30,43)(31,44)(32,37)(33,41)(35,39)(40,42);(2,4)(13,16,19,30)(14,21,17,31)(15,23,27,41)(18,33,34,29)(20,35,24,42)(22,40,32,39)(25,28,26,43)(36,38,37,44);(6,9)(10,12)(13,17,19,14)(15,24,27,20)(16,21,30,31)(18,32,34,22)(23,35,41,42)(25,37,26,36)(28,38,43,44)(29,39,33,40);(5,7,8,11)(6,10,9,12)(15,25)(18,34)(20,36)(22,32)(23,28)(24,37)(26,27)(29,33)(35,38)(39,40)(41,43)(42,44)

Polycyclic presentation:

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

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, 6579952);