#360558

order	512
nilpotency class	3
isoclinism	isoclinic class of #222256
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2189 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)(5,6,10,17,11,15,9,16)(7,18,22,29,23,12,21,30)(8,13,24,32,25,14,26,33)(19,34,20,28,35,27,36,31);(5,7)(6,12)(8,20)(9,22)(10,21)(11,23)(13,31)(14,28)(15,18)(16,30)(17,29)(19,24)(25,36)(26,35)(27,32)(33,34);(1,2,3,4)(5,8)(6,13)(7,19)(9,26)(10,24)(11,25)(12,27)(14,15)(16,33)(17,32)(18,34)(20,22)(21,36)(23,35)(28,29)(30,31);(5,8)(6,14)(7,20)(9,24)(10,26)(11,25)(12,28)(13,15)(16,33)(17,32)(18,31)(19,22)(21,35)(23,36)(27,29)(30,34)

Polycyclic presentation:

2^9|a^2=h,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^d=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, 267754);