#2601675

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1849 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)(16,24,26,20)(23,27,25,28)(29,30)(31,36)(32,39)(33,35)(34,41)(37,42)(38,44)(40,43);(7,11)(9,12)(14,18)(15,22)(20,24)(23,25)(29,31,35,32)(30,36,33,39)(34,38,42,40)(37,43,41,44);(5,7)(6,9)(8,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28)(29,32,35,31)(30,36,33,39)(34,40,42,38)(37,43,41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,22)(14,19,18,21)(16,25,26,23)(20,28,24,27)(29,33,35,30)(31,36,32,39)(34,41,42,37)(38,43,40,44);(13,16,17,26)(14,20,18,24)(15,23,22,25)(19,27,21,28)(29,34)(30,37)(31,38)(32,40)(33,41)(35,42)(36,43)(39,44)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=fg,d^2=g,e^2=f,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, 2508871);