#586768

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

1974 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)(5,6,9,10)(7,11,12,8)(13,14)(15,19)(16,17)(18,20)(21,22,24,26)(23,25,28,27);(5,7,9,12)(6,8,10,11)(14,16)(15,20)(21,22)(23,27)(24,26)(25,28);(6,10)(8,11)(13,15,17,20)(14,18,16,19)(21,22,24,26)(23,25,28,27);(5,8)(6,7)(9,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28);(5,7)(6,8)(9,12)(10,11)(13,16,17,14)(15,19,20,18)(21,23)(22,25)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=h,e^2=fgh,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, 493964);