#6443698

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350894);