#392792

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1888 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ei,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,d^b=di,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299988);