#582228

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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

1962 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)(5,6,8,10)(7,11,12,9)(14,19)(15,17)(23,27)(26,28);(5,6)(7,9)(8,10)(11,12)(13,14,18,19)(15,16,17,20)(22,25)(26,28);(5,6,8,10)(7,9,12,11)(13,15,18,17)(14,20,19,16)(21,22,24,25)(23,26,27,28);(5,6)(7,9)(8,10)(11,12)(13,16,18,20)(14,15,19,17)(21,23)(22,26)(24,27)(25,28);(5,7,8,12)(6,9,10,11)(13,17,18,15)(14,20,19,16)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 489424);