#4602126

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(6,10)(7,11)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30,35,39)(31,37,32,36)(33,34,44,38)(40,41,42,43);(5,6)(7,8)(9,10)(11,12)(15,22)(18,21)(23,28)(25,27)(29,31,35,32)(30,36,39,37)(33,42,44,40)(34,41,38,43);(6,10)(8,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,32,35,31)(30,37,39,36)(33,42,44,40)(34,41,38,43);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,15,17,22)(14,18,20,21)(16,23,24,28)(19,25,26,27)(29,33)(30,38)(31,40)(32,42)(34,39)(35,44)(36,41)(37,43);(5,8)(6,7)(9,12)(10,11)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,34,35,38)(30,33,39,44)(31,41,32,43)(36,42,37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fh,d^2=fgh,e^2=g,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, 4509322);