#2530441

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2091 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)(4,6)(7,8)(9,10)(11,16)(12,14)(13,15)(19,23)(22,24)(25,26,28,30)(27,31,32,29);(2,3)(6,7)(9,11,14,13)(10,15,12,16)(25,27)(26,29)(28,32)(30,31);(4,8)(6,7)(11,13)(15,16)(17,18,21,20)(19,22,23,24)(25,27,28,32)(26,29,30,31);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,13,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,29)(28,30)(31,32);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,15,12,16)(17,20,21,18)(19,24,23,22)(25,27)(26,29)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fgi,d^2=fg,e^2=fghi,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, 2437637);