#4757451

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1758 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)(11,16)(12,13)(14,15)(17,18,21,22)(19,23,24,20)(25,26)(27,33)(28,30)(29,36)(31,35)(32,38)(34,39)(37,40);(9,11,14,13)(10,12,15,16)(17,19)(18,20)(21,24)(22,23)(26,28)(31,33)(32,36)(39,40);(10,15)(11,13)(19,24)(20,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,12,15,16)(17,20)(18,19)(21,23)(22,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4664647);