#2244969

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1809 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)(3,7)(4,6)(5,8)(10,13)(11,15)(17,18,23,21)(19,24,22,20)(25,26,28,30)(27,31,32,29);(1,3,6,5)(2,7,4,8)(11,15)(14,16)(17,19,23,22)(18,24,21,20)(25,26)(27,29)(28,30)(31,32);(1,2,6,4)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,22)(19,21)(23,24)(25,26)(27,31)(28,30)(29,32);(1,4,6,2)(3,7,5,8)(9,11,12,15)(10,14,13,16)(17,21,23,18)(19,24,22,20);(1,5,6,3)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,22,23,19)(18,24,21,20)(25,27,28,32)(26,29,30,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=fghi,d^2=fg,e^2=fh,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, 2152165);