#6682775

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2247 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,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, 6589971);