#6445134

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2032 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)(13,14,17,21)(15,23,25,19)(16,27,29,43)(18,31,28,22)(20,34,35,30)(24,38,40,42)(26,33,39,37)(32,41,36,44);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,24)(17,25)(18,26)(20,32)(21,23)(22,33)(27,42)(28,39)(29,40)(30,41)(31,37)(34,44)(35,36)(38,43);(2,4)(13,16,28,34,17,29,18,30)(14,20,31,43,21,35,22,27)(15,24,39,44,25,40,26,41)(19,32,37,38,23,36,33,42);(2,4)(15,25)(16,29)(19,23)(20,35)(26,39)(27,43)(30,34)(33,37);(5,7,8,11)(6,10,9,12)(15,25)(19,23)(24,40)(26,39)(32,36)(33,37)(38,42)(41,44)

Polycyclic presentation:

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

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, 6352330);