#552695

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(10,12)(14,15)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31);(2,4)(5,6,7,8)(9,10,13,12)(11,15,16,14)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31);(1,2)(3,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32);(9,11,13,16)(10,14,12,15)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30);(9,12,13,10)(11,15,16,14)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=g,d^2=f,e^2=f,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, 459891);