#358918

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=gi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,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, 266114);