#2244963

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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=fghi,d^2=g,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, 2152159);