#5213497

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2056 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)(5,6,8,10)(7,9,11,12)(13,14,18,15)(16,21,23,19)(17,20,25,22)(24,27,28,26)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,7,8,11)(6,9,10,12)(14,15)(16,23)(20,22)(24,28)(31,38)(33,41)(35,37)(36,40);(5,7)(6,9)(8,11)(10,12)(13,15,18,14)(16,19,23,21)(17,20,25,22)(24,27,28,26)(29,31,34,38)(30,35,32,37)(33,39,41,44)(36,42,40,43);(1,2)(3,4)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,32,34,30)(31,35,38,37)(33,40,41,36)(39,42,44,43);(5,6)(7,9)(8,10)(11,12)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,33,34,41)(30,36,32,40)(31,39,38,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=fg,d^2=fg,e^2=fg,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, 5120693);