#354561

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fi,d^2=gi,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^c=di

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, 261757);