#358274

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×#393
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1686 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)(7,17,19,29,21,12,20,28)(8,22,25,31,24,32,26,13)(18,34,36,33,35,30,23,27);(1,2)(3,4)(5,7,9,19,11,21,10,20)(6,12,15,28,14,17,16,29)(8,23,25,35,24,36,26,18)(13,30,22,33,31,34,32,27);(2,4)(5,8)(6,13,14,31)(7,18)(9,25)(10,26)(11,24)(12,27,17,33)(15,22,16,32)(19,23)(20,36)(21,35)(28,30,29,34);(8,24)(13,31)(18,35)(22,32)(23,36)(25,26)(27,33)(30,34)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=di,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, 265470);