#6440191

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

2138 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)(13,14,18,23)(15,24,27,20)(16,29,31,37)(17,22,19,33)(21,30,36,32)(25,40,42,44)(26,39,28,35)(34,43,38,41);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,25)(17,26)(18,27)(19,28)(21,34)(22,35)(23,24)(29,44)(30,41)(31,42)(32,43)(33,39)(36,38)(37,40);(2,4)(13,16,18,31)(14,21,23,36)(15,25,27,42)(17,32,19,30)(20,34,24,38)(22,37,33,29)(26,43,28,41)(35,40,39,44);(2,4)(13,17,18,19)(14,22,23,33)(15,26,27,28)(16,30,31,32)(20,35,24,39)(21,29,36,37)(25,41,42,43)(34,44,38,40);(5,7,8,11)(6,10,9,12)(15,27)(20,24)(25,42)(26,28)(34,38)(35,39)(40,44)(41,43)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,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,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, 6347387);