#545868

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2238 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)(5,6,8,10)(7,9,11,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,27)(24,26)(25,28);(5,6)(7,9)(8,10)(11,12)(14,17)(18,19)(21,22,24,26)(23,25,28,27);(1,3,4,2)(7,11)(9,12)(13,14,16,17)(15,18,20,19)(21,22,24,26)(23,25,28,27);(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28);(5,6)(7,9)(8,10)(11,12)(13,15,16,20)(14,18,17,19)(21,23)(22,25)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fi,d^2=fg,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, 453064);