#2549019

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

1614 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=fhi,d^2=fh,e^2=fghi,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, 2456215);