#6439995

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2164 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)(16,29,31,36)(17,22,19,32)(21,33,37,30)(25,40,42,44)(26,39,28,35)(34,41,38,43);(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)(32,39)(33,43)(36,40)(37,38);(2,4)(13,16)(14,21)(15,25)(17,30)(18,31)(19,33)(20,34)(22,36)(23,37)(24,38)(26,41)(27,42)(28,43)(29,32)(35,40)(39,44);(2,4)(13,17,18,19)(14,22,23,32)(15,26,27,28)(16,30,31,33)(20,35,24,39)(21,36,37,29)(25,41,42,43)(34,40,38,44);(5,7,8,11)(6,10,9,12)(15,27)(16,31)(20,24)(21,37)(26,28)(29,36)(30,33)(35,39)

Polycyclic presentation:

2^9|a^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,e^c=ei,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347191);