#6552775

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1817 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)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,29)(21,32)(26,34)(27,35)(28,36)(31,38)(33,37)(39,42)(40,43)(41,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,29,19,28,17,27)(14,20,22,25,24,35,23,36)(16,26,31,40,33,41,32,42)(21,34,37,39,38,44,30,43);(2,4)(6,9)(10,12)(13,16,17,32,19,33,18,31)(14,21,23,30,24,38,22,37)(15,26,27,42,28,41,29,40)(20,34,36,43,35,44,25,39);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,29)(16,31,33,32)(20,25,35,36)(21,30,38,37)(26,40,41,42)(34,43,44,39);(7,11)(10,12)(15,28)(20,35)(25,36)(26,41)(27,29)(34,44)(39,43)(40,42)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459971);