#2228034

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1724 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,13)(14,15)(19,23)(22,24)(25,26,29,27)(28,31,32,30);(1,3,6,5)(2,7,4,8)(9,10,14,15)(11,16,12,13)(18,21)(22,24);(3,5)(7,8)(9,11,14,12)(10,13,15,16)(17,18,20,21)(19,22,23,24)(25,27,29,26)(28,31,32,30);(1,4,6,2)(3,7,5,8)(9,12,14,11)(10,13,15,16)(17,19)(18,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32);(1,5,6,3)(2,7,4,8)(9,13)(10,11)(12,15)(14,16)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=fi,d^2=f,e^2=fg,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, 2135230);