#4606510

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1865 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=fg,e^2=f,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, 4513706);