#358278

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1582 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,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,e^b=ei,g^c=gi,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, 265474);