#359291

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=eh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266487);