#392972

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×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

2116 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:

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

Polycyclic presentation:

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

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, 300168);