#6668268

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#36078	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2686 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6575464);