#2072353

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

1585 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,13)(14,15)(17,18,21,22)(19,20,24,23)(26,30)(31,33)(32,36)(38,39);(2,6)(4,7)(10,15)(13,16)(18,22)(20,23)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,38,40,39);(9,11,14,12)(10,16,15,13)(17,18,21,22)(19,23,24,20)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,11)(10,13,15,16)(17,19)(18,23)(20,22)(21,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,4)(2,3)(5,7)(6,8)(9,13)(10,11)(12,15)(14,16)(17,20,21,23)(18,19,22,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fhi,d^2=fi,e^2=gh,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, 1979549);