#392189

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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

3402 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)(5,6,10,14,11,15,9,24)(7,16,20,27,18,31,19,12)(8,21,22,28,23,13,25,29)(17,32,34,36,33,30,35,26);(5,7,9,19,11,18,10,20)(6,12,14,16,15,27,24,31)(8,17,22,34,23,33,25,35)(13,26,28,30,21,36,29,32);(5,8)(6,13)(7,17)(9,22)(10,25)(11,23)(12,26)(14,28)(15,21)(16,30)(18,33)(19,34)(20,35)(24,29)(27,36)(31,32);(2,4)(7,18)(12,27)(16,31)(17,33)(19,20)(26,36)(30,32)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=e,e^2=i,h^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, 299385);