#545520

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

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

2570 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=ghi,d^2=g,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, 452716);