#2805611

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
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

1557 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,c^2=fgh,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2712807);