#357030

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1246 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=efgi,d^2=fgi,f^2=i,g^2=i,b^a=be,c^a=cf,d^b=dei,f^c=fi,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264226);