#6544782

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1992 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)(13,14,17,21,18,22,30,36)(15,19,25,33,24,32,40,43)(16,26,27,34,28,35,29,20)(23,37,39,42,38,44,41,31);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,19,22,32)(16,23,28,38)(17,25,30,40)(20,31,34,42)(21,33,36,43)(26,37,35,44)(27,39,29,41);(6,9)(10,12)(13,16,18,28)(14,20,22,34)(15,23,24,38)(17,29,30,27)(19,31,32,42)(21,35,36,26)(25,41,40,39)(33,44,43,37);(5,7)(6,10)(8,11)(9,12)(15,24)(19,32)(23,38)(25,40)(31,42)(33,43)(37,44)(39,41);(7,11)(10,12)(15,24)(19,32)(23,38)(25,40)(31,42)(33,43)(37,44)(39,41)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451978);