#556393

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,d^b=dg,e^b=ef,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, 463589);