#556390

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2914 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 463586);