#4422262

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

1856 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)(7,9,11,12)(14,18)(16,23)(19,25)(21,24)(33,41)(36,40)(38,44)(42,43);(13,14,17,18)(15,19,22,25)(16,20,23,26)(21,27,24,28)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,26)(24,28)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,22)(14,19,18,25)(16,24,23,21)(20,28,26,27)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(5,7,8,11)(6,9,10,12)(13,16,17,23)(14,20,18,26)(15,21,22,24)(19,27,25,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fh,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4329458);