#517618

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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

4578 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 424814);