#6542532

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×#3541	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3008 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)(13,14,19,33,18,23,17,22)(15,24,26,20,28,35,27,36)(16,29,30,38,32,21,31,37)(25,40,42,44,43,39,41,34);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,26)(18,28)(19,27)(21,34)(22,35)(23,36)(24,33)(29,44)(30,41)(31,42)(32,43)(37,39)(38,40);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,31)(18,32)(19,30)(20,34)(22,38)(23,29)(24,39)(26,42)(27,41)(28,43)(33,37)(35,40)(36,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,23,33)(15,26,28,27)(16,30,32,31)(20,35,36,24)(21,37,29,38)(25,41,43,42)(34,39,44,40);(7,11)(10,12)(13,17,18,19)(14,22,23,33)(15,26,28,27)(16,31,32,30)(20,35,36,24)(21,38,29,37)(25,42,43,41)(34,40,44,39)

Polycyclic presentation:

2^9|a^2=fg,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,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, 6449728);