#4861981

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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

1920 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)(7,11)(9,12)(13,14)(15,21)(16,18)(17,20)(19,23)(22,27)(24,25)(26,28)(29,30,35,32)(31,38,33,36)(34,40,42,37)(39,43,41,44);(5,6,8,10)(7,9,11,12)(14,16)(19,21)(20,24)(26,27)(29,31,35,33)(30,36,32,38)(34,39,42,41)(37,43,40,44);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28)(29,32,35,30)(31,36,33,38)(34,40,42,37)(39,43,41,44);(1,2)(3,4)(13,16,18,14)(15,21,23,19)(17,20,25,24)(22,26,28,27)(29,33,35,31)(30,36,32,38)(34,39,42,41)(37,44,40,43);(5,7)(6,9)(8,11)(10,12)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,34,35,42)(30,37,32,40)(31,39,33,41)(36,43,38,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=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, 4769177);