#4603790

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
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

1919 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)(6,9)(7,11)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30,35,40)(31,37,32,36)(33,38,39,34)(41,44,43,42);(6,9)(10,12)(15,22)(18,21)(23,28)(25,27)(29,31,35,32)(30,36,40,37)(33,43,39,41)(34,42,38,44);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,32,35,31)(30,37,40,36)(33,43,39,41)(34,42,38,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,23)(17,22)(19,25)(20,21)(24,28)(26,27)(29,33,35,39)(30,38,40,34)(31,41,32,43)(36,42,37,44);(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,34,35,38)(30,39,40,33)(31,42,32,44)(36,41,37,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=f,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,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, 4510986);