#395574

order	512
nilpotency class	3
isoclinism	isoclinic class of #393900
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^c=ei,f^c=fi,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, 302770);