#516474

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

2818 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)(5,6,10,8)(7,12,9,11)(14,18)(16,19)(22,27)(23,25);(5,7,10,9)(6,11,8,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27);(6,8)(11,12)(13,15,17,20)(14,19,18,16)(21,23,26,25)(22,28,27,24);(5,8,10,6)(7,12,9,11)(13,15)(14,16)(17,20)(18,19)(21,24,26,28)(22,23,27,25);(5,9,10,7)(6,12,8,11)(13,16)(14,15)(17,19)(18,20)(21,25,26,23)(22,28,27,24)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=h,d^2=g,e^2=g,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, 423670);