#516661

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

2198 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 423857);