#2415135

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2099 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)(9,10,13,11)(12,14,16,15)(18,19)(20,24)(25,26,30,32)(27,29,28,31);(1,2,4,6)(3,5,7,8)(10,11)(14,15)(17,18,21,19)(20,22,24,23)(25,26)(27,29)(28,31)(30,32);(3,7)(5,8)(9,10,13,11)(12,15,16,14)(17,18,21,19)(20,23,24,22)(25,27,30,28)(26,31,32,29);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19,21,18)(20,23,24,22)(25,28,30,27)(26,31,32,29);(1,3)(2,5)(4,7)(6,8)(9,12,13,16)(10,14,11,15)(17,20)(18,22)(19,23)(21,24)(25,29,30,31)(26,27,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=hi,d^2=hi,e^2=h,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, 2322331);