#545513

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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3182 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)(10,13)(14,16)(18,22)(20,28)(23,25)(26,32);(2,4)(10,13)(11,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,2,3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(9,11,12,15)(10,14,13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 452709);