#379693

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

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

848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=fgh,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, 286889);