#4600732

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 2097152
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

1780 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)(6,9)(7,8)(13,14)(15,18)(16,21)(17,20)(19,23)(22,25)(24,27)(26,28)(29,30,34,35)(31,36,38,32)(33,37,42,43)(39,44,41,40);(5,6,10,9)(7,11,8,12)(14,15)(16,23)(20,22)(24,28)(30,35)(31,38)(37,43)(39,41);(5,7,10,8)(6,11,9,12)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,26,28,27)(29,31)(30,36)(32,35)(33,41)(34,38)(37,40)(39,42)(43,44);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28)(29,32,34,36)(30,31,35,38)(33,40,42,44)(37,39,43,41);(5,9,10,6)(7,11,8,12)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,33,34,42)(30,37,35,43)(31,39,38,41)(32,40,36,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=g,d^2=gh,e^2=f,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, 4507928);