#373217

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1000 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)(7,17,21,12)(8,22)(9,25,27,14)(10,16)(11,29)(13,24)(15,28)(18,34,26,32)(19,31)(20,30,35,33)(23,36);(2,4)(5,7,10,21)(6,12)(8,18,23,26)(9,19,27,24)(11,20,28,35)(13,14)(15,30)(16,17)(22,32)(25,31)(29,33)(34,36);(1,2,3,4)(5,8,11,24,10,23,28,19)(6,13,15,22,16,31,29,36)(7,18,20,9,21,26,35,27)(12,14,30,32,17,25,33,34);(5,9,11,26,10,27,28,18)(6,14,15,32,16,25,29,34)(7,19,20,8,21,24,35,23)(12,13,30,22,17,31,33,36)

Polycyclic presentation:

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

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, 280413);