#6558174

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2476 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)(13,14,18,22)(15,23,27,39)(16,20,29,36)(17,31,24,21)(19,32,34,26)(25,38,28,42)(30,44,40,37)(33,43,35,41);(5,6)(7,10)(8,9)(11,12)(13,15,24,32,18,27,17,26)(14,19,31,39,22,34,21,23)(16,28,40,41,29,25,30,43)(20,35,44,38,36,33,37,42);(1,2,3,4)(6,9)(10,12)(13,16,18,29)(14,20,22,36)(15,25,27,28)(17,30,24,40)(19,33,34,35)(21,37,31,44)(23,38,39,42)(26,41,32,43);(5,7)(6,10)(8,11)(9,12)(16,29)(20,36)(25,28)(30,40)(33,35)(37,44)(38,42)(41,43);(7,11)(10,12)(16,29)(20,36)(25,28)(30,40)(33,35)(37,44)(38,42)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^c=ei,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465370);