#6667817

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2715 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)(7,11)(8,10)(9,12)(16,27)(20,36)(24,40)(28,38)(29,32)(34,43)(39,44)(41,42);(5,7)(6,9)(8,12)(10,11)(13,14,18,22)(15,23,26,37)(16,28,27,38)(17,30,33,21)(19,31,35,25)(20,32,36,29)(24,34,40,43)(39,42,44,41);(2,4)(13,15,18,26)(14,19,22,35)(16,24,27,40)(17,31,33,25)(20,34,36,43)(21,37,30,23)(28,39,38,44)(29,42,32,41);(6,10)(9,11)(16,27)(20,36)(24,40)(28,38)(29,32)(34,43)(39,44)(41,42);(7,12)(9,11)(13,16,18,27)(14,20,22,36)(15,24,26,40)(17,32,33,29)(19,34,35,43)(21,38,30,28)(23,39,37,44)(25,42,31,41)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^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,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, 6575013);