#356451

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1362 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)(6,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22,25,23,24,26,28,27);(5,6)(7,11)(8,12)(9,10)(13,15)(14,17)(16,20)(18,19)(21,23,25,26,24,27,28,22);(5,7,9,8)(6,11,10,12)(14,18)(17,19)(22,26)(23,27);(5,8,9,7)(6,11,10,12)(15,20)(17,19)(22,26)(23,27)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263647);