#397963

order	512
nilpotency class	3
isoclinism	isoclinic class of #394353
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305159);