#380195

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

1026 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:

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

Polycyclic presentation:

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

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, 287391);