#2826562

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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

1860 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=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, 2733758);