#544503

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,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, 451699);