#6682621

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#39335	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2234 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,23,25,39)(16,20,27,35)(17,30,31,21)(19,32,34,26)(24,38,28,42)(29,44,40,37)(33,43,36,41);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(16,27)(20,35)(24,28)(29,40)(33,36)(37,44)(38,42)(41,43);(2,4)(13,15,17,26,18,25,31,32)(14,19,21,23,22,34,30,39)(16,28,29,43,27,24,40,41)(20,36,37,42,35,33,44,38);(6,10)(9,11)(13,16,18,27)(14,20,22,35)(15,24,25,28)(17,29,31,40)(19,33,34,36)(21,37,30,44)(23,38,39,42)(26,41,32,43);(2,4)(7,12)(9,11)(15,25)(16,27)(19,34)(20,35)(23,39)(26,32)(29,40)(37,44)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^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,d^c=di,e^c=ei,e^d=ei

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, 6589817);