#362612

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1367 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,h^c=hi,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, 269808);