#6542552

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

3032 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)(13,14,19,32,18,23,17,22)(15,24,26,20,28,34,27,35)(16,21,29,36,31,38,30,37)(25,39,41,33,42,43,40,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,26)(18,28)(19,27)(21,33)(22,34)(23,35)(24,32)(29,40)(30,41)(31,42)(36,39)(37,43)(38,44);(2,4)(6,9)(10,12)(13,16,18,31)(14,21,23,38)(15,25,28,42)(17,30,19,29)(20,33,35,44)(22,37,32,36)(24,39,34,43)(26,41,27,40);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,23,32)(15,26,28,27)(16,29,31,30)(20,34,35,24)(21,36,38,37)(25,40,42,41)(33,39,44,43);(7,11)(10,12)(13,17,18,19)(14,22,23,32)(15,26,28,27)(16,30,31,29)(20,34,35,24)(21,37,38,36)(25,41,42,40)(33,43,44,39)

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,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, 6449748);