#536395

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

2406 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)(7,10)(8,9)(11,12)(13,14,17,19)(15,20,16,18)(21,22,25,28)(23,24,27,26);(1,3,4,2)(6,8)(10,12)(13,15,17,16)(14,18,19,20);(7,11)(10,12)(15,16)(18,20)(21,23,25,27)(22,26,28,24);(5,7,9,11)(6,10,8,12)(13,16,17,15)(14,18,19,20)(21,22)(23,26)(24,27)(25,28);(5,8,9,6)(7,12,11,10)(13,14)(15,20)(16,18)(17,19)(21,24,25,26)(22,27,28,23)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 443591);