#2273737

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1708 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=fh,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, 2180933);