#545319

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

1842 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 452515);