#2076470

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
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

1782 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fghi,d^2=fhi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=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, 1983666);