#582622

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2634 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gi,d^2=g,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, 489818);