#2636744

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,6,7)(3,8,4,5)(9,10)(11,12)(13,16)(14,15)(18,20)(19,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(2,7)(3,4)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,21)(22,24)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(1,3,6,4)(2,5,7,8)(9,12)(10,15)(11,16)(13,14)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,4,6,3)(2,5,7,8)(9,12)(10,11)(13,14)(15,16)(17,20,21,18)(19,23,24,22)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,5)(2,4)(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,c^2=hi,d^2=h,e^2=fg,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.

Relationships

maximal subgroup of	#72267719
maximal normal subgroup of	#72267719
maximal quotient of	#72267734

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2543940);