#4962572

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1819 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4869768);