#2548969

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

1582 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=fhi,d^2=fh,e^2=ghi,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, 2456165);