#6440174

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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

2140 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)(15,20)(16,28)(17,22)(18,32)(19,23)(21,30)(24,38)(25,34)(26,35)(27,42)(29,36)(31,37)(33,41)(39,43)(40,44);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,24)(17,25)(18,27)(19,26)(21,33)(22,34)(23,35)(28,38)(29,39)(30,41)(31,40)(32,42)(36,43)(37,44);(2,4)(13,16,19,31)(14,21,23,36)(15,24,26,40)(17,30,18,29)(20,33,35,43)(22,37,32,28)(25,41,27,39)(34,44,42,38);(2,4)(13,17,19,18)(14,22,23,32)(15,25,26,27)(16,29,31,30)(20,34,35,42)(21,28,36,37)(24,39,40,41)(33,38,43,44);(5,7,8,11)(6,10,9,12)(15,26)(20,35)(24,40)(25,27)(33,43)(34,42)(38,44)(39,41)

Polycyclic presentation:

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

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, 6347370);