#536394

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	C₂×#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2742 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)(13,14,16,19)(15,18,20,17)(21,22,25,28)(23,24,27,26);(1,3,4,2)(6,10)(8,11)(13,15)(14,17)(16,20)(18,19);(6,10)(7,12)(13,15,16,20)(14,18,19,17)(21,23,25,27)(22,26,28,24);(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,27)(25,28);(5,8)(6,7)(9,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,24,25,26)(22,27,28,23)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fg,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, 443590);