#365705

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ef,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di

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, 272901);