#358033

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1734 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,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, 265229);