#360590

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2148 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)(5,6,9,15,11,16,10,14)(7,17,20,27,21,32,19,12)(8,13,23,30,24,31,25,29)(18,33,34,36,35,28,22,26);(5,7)(6,12)(8,22)(9,20)(10,19)(11,21)(13,28)(14,32)(15,17)(16,27)(18,23)(24,34)(25,35)(26,30)(29,36)(31,33);(1,2,3,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,29)(15,30)(16,31)(17,33)(19,22)(20,34)(21,35)(27,36)(28,32);(5,6)(7,12)(8,13)(9,15)(10,14)(11,16)(17,20)(18,26)(19,32)(21,27)(22,28)(23,30)(24,31)(25,29)(33,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267786);