#2472953

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1772 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=fg,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, 2380149);