#2601580

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1974 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)(15,21,23,19)(16,24,25,20)(22,26,28,27)(29,30)(31,35)(32,38)(33,41)(34,37)(36,42)(39,44)(40,43);(7,11)(9,12)(14,18)(19,21)(20,24)(26,27)(29,31,34,32)(30,35,37,38)(33,39,42,40)(36,43,41,44);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28)(29,30)(31,38)(32,35)(33,36)(34,37)(39,44)(40,43)(41,42);(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fg,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, 2508776);