#555911

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4602 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef

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, 463107);