#582492

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

2392 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)(8,12)(9,10)(13,14)(15,20)(16,17)(18,19)(21,22)(23,26)(24,28)(25,27);(1,3,4,2)(6,10)(7,12)(13,15)(14,17)(16,18)(19,20)(21,23)(22,24)(25,28)(26,27);(6,10)(7,12)(14,19)(15,16)(21,23,25,28)(22,26,27,24);(5,7)(6,8)(9,12)(10,11)(13,16,18,15)(14,17,19,20)(21,23)(22,24)(25,28)(26,27);(5,8)(6,7)(9,11)(10,12)(13,17)(14,15)(16,19)(18,20)(21,24)(22,23)(25,26)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fgh,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, 489688);