#576755

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

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, 483951);