#587552

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2296 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)(7,11)(8,10)(9,12)(13,14,16,18)(15,17,19,20)(21,22,24,26)(23,25,27,28);(1,3,4,2)(5,6,8,10)(7,11,12,9)(15,19)(17,20)(23,27)(25,28);(5,6,8,10)(7,9,12,11)(13,14)(15,17)(16,18)(19,20)(21,23)(22,25)(24,27)(26,28);(5,6)(7,9)(8,10)(11,12)(13,15,16,19)(14,17,18,20)(21,22)(23,25)(24,26)(27,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,17)(16,18)(19,20)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=f,d^2=fg,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 494748);