#360162

order	512
nilpotency class	3
isoclinism	isoclinic class of #196278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 267358);