#2601764

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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

1860 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)(13,14,17,18)(15,21,22,19)(16,24,25,20)(23,26,28,27)(29,30)(31,34)(32,40)(33,37)(35,39)(36,41)(38,43)(42,44);(7,11)(9,12)(14,18)(15,22)(20,24)(23,28)(29,31,33,39)(30,34,37,35)(32,38,41,44)(36,42,40,43);(5,7)(6,9)(8,11)(10,12)(13,15,17,22)(14,19,18,21)(16,23,25,28)(20,26,24,27)(29,31)(30,35)(32,38)(33,39)(34,37)(36,43)(40,42)(41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,22)(23,27)(26,28)(29,30)(31,35)(32,36)(33,37)(34,39)(38,43)(40,41)(42,44);(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,32,33,41)(30,36,37,40)(31,38,39,44)(34,42,35,43)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2508960);