#576784

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=h,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, 483980);