#2607035

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2527 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2,3,4)(6,8)(7,12)(14,17)(18,19)(21,22,27,30)(23,28,24,33)(25,36,29,26)(31,32,34,35);(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,17)(18,20)(21,23,27,24)(22,28,30,33)(25,34,29,31)(26,32,36,35);(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,28,30,33)(25,34,29,31)(26,35,36,32);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,15,16,20)(14,18,17,19)(21,25,27,29)(22,26,30,36)(23,31,24,34)(28,35,33,32);(5,7)(6,11)(8,10)(9,12)(13,14)(15,19)(16,17)(18,20)(21,26)(22,29)(23,32)(24,35)(25,30)(27,36)(28,31)(33,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=fg,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,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, 2514231);