#5216838

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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

1702 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)(5,6,9,11)(7,8,12,10)(13,14,18,15)(16,21,23,19)(17,20,25,22)(24,27,28,26)(29,30)(31,37)(32,36)(33,34)(35,38)(39,44)(40,42)(41,43);(5,6,9,11)(7,8,12,10)(14,15)(16,23)(20,22)(24,28)(30,34)(31,38)(32,40)(43,44);(5,7,9,12)(6,10,11,8)(13,15,18,14)(16,19,23,21)(17,20,25,22)(24,27,28,26)(29,31)(30,35)(32,41)(33,38)(34,37)(36,44)(39,40)(42,43);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,42)(41,44);(5,8)(6,7)(9,10)(11,12)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,32,33,40)(30,36,34,42)(31,39,38,41)(35,43,37,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fg,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 5124034);