#111427

order	512
nilpotency class	3
isoclinism	isoclinic class of #102079
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18623);