#554192

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

2666 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fgi,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, 461388);