#6542568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3539	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,d^c=di,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, 6449764);