#4610468

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

1846 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)(5,6,8,10)(7,11,12,9)(14,18)(16,23)(19,24)(22,27)(30,31)(33,41)(35,37)(39,44);(5,7)(6,9)(8,12)(10,11)(14,18)(15,21)(20,25)(22,27)(29,30,34,31)(32,37,40,35)(33,36,41,38)(39,43,44,42);(5,6,8,10)(7,9,12,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,30)(32,37,40,35)(33,36,41,38)(39,42,44,43);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,35)(31,37)(33,39)(34,40)(36,42)(38,43)(41,44);(5,6)(7,9)(8,10)(11,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,41)(30,36,31,38)(32,39,40,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fh,d^2=h,e^2=g,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, 4517664);