#358260

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

1578 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,h^b=hi,f^c=fi,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, 265456);