#2280890

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

1716 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)(3,4)(5,7)(6,8)(9,10,14,16)(11,13,12,15)(18,21)(19,23)(26,30)(28,32);(1,3,5,8)(2,6,7,4)(9,10)(11,13)(12,15)(14,16)(17,18)(19,22)(20,21)(23,24)(26,30)(27,31);(3,8)(4,6)(9,11,14,12)(10,15,16,13)(17,19,20,23)(18,22,21,24)(25,26)(27,28)(29,30)(31,32);(1,2)(3,4)(5,7)(6,8)(9,12,14,11)(10,15,16,13)(17,18)(19,22)(20,21)(23,24)(25,27)(26,28)(29,31)(30,32);(1,4)(2,3)(5,6)(7,8)(9,13,14,15)(10,11,16,12)(17,19)(18,22)(20,23)(21,24)(25,28)(26,27)(29,32)(30,31)

Polycyclic presentation:

2^9|a^2=fgh,b^2=ghi,c^2=fghi,d^2=fgh,e^2=fgh,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, 2188086);