#392913

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1634 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)(5,6)(7,8)(9,10,12,11)(15,21)(16,23)(18,25)(20,26)(22,27)(24,28);(2,4)(9,11,12,10)(13,14)(15,22)(16,20)(17,19)(18,24)(21,27)(23,26)(25,28);(13,15,23,28,17,24,16,21)(14,18,26,27,19,22,20,25);(6,8)(14,19)(18,22)(20,26)(25,27)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fi,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,c^b=ci,d^b=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, 300109);