#392187

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3442 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,d^b=di,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299383);