#950613

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

1744 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,20,22)(19,23,24,21)(26,30)(27,32);(2,3)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24)(26,30)(27,32);(2,3)(6,7)(10,12)(11,16)(17,18,20,22)(19,21,24,23)(25,26,29,30)(27,31,32,28);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,30)(29,32);(1,4,5,8)(2,6,3,7)(9,13)(10,11)(12,16)(14,15)(17,19)(18,21)(20,24)(22,23)(25,28,29,31)(26,27,30,32)

Polycyclic presentation:

2^9|a^2=gi,c^2=fg,d^2=fgi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 857809);