#6445624

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2200 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352820);