#577624

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2504 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)(9,10,13,14)(11,16,15,12)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32);(6,8)(10,14)(11,15)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32);(5,6)(7,8)(10,14)(12,16)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32);(1,3,4,2)(9,11,13,15)(10,12,14,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32);(9,12)(10,15)(11,14)(13,16)(17,22,23,18)(19,26,20,24)(21,29,30,25)(27,32,28,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=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, 484820);