#2199589

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1924 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2106785);