#391622

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
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

816 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)(3,5)(4,6)(7,8)(10,11)(14,15)(18,21)(22,23)(25,26,28,30,27,29,31,32);(1,3)(2,5)(4,7)(6,8)(9,10,13,11)(12,14,16,15)(17,18,20,21)(19,23,24,22);(2,6)(5,8)(9,11,13,10)(12,14,16,15)(17,19,20,24)(18,22,21,23)(26,29)(30,32);(3,7)(5,8)(9,12,13,16)(10,14,11,15)(18,21)(19,24)(26,29)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=defi,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298818);