#358258

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×#394
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1618 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)(7,17,21,29)(8,22,26,32)(9,15)(10,16)(11,14)(12,19,28,20)(13,25,31,24)(18,33)(23,36)(27,34)(30,35);(5,7,9,20,11,21,10,19)(6,12,15,29,14,28,16,17)(8,23,25,34,26,35,24,18)(13,30,22,33,31,36,32,27);(5,8,9,24,11,26,10,25)(6,13,15,32,14,31,16,22)(7,18,19,23,21,34,20,35)(12,27,17,30,28,33,29,36);(6,14)(12,28)(13,31)(15,16)(17,29)(22,32)(27,33)(30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,e^2=i,f^2=i,g^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, 265454);