#4973731

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

1650 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)(4,6,8,7)(10,13)(11,15)(17,18,22,24)(19,20,21,23)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(9,10,12,13)(11,14,15,16)(26,27)(29,37)(31,33)(35,40);(2,3)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,21)(18,23,24,20)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,22,23)(18,19,24,21)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,21,22,19)(18,23,24,20)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fi,d^2=fgh,e^2=fgh,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, 4880927);