#6445114

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2018 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352310);