#372564

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1231 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,d^2=fgi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,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, 279760);