#4471686

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1962 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)(29,30,34,38)(31,36,32,35)(33,37,41,44)(39,43,40,42);(6,9)(10,12)(15,22)(18,21)(23,28)(25,27)(29,31,34,32)(30,35,38,36)(33,40,41,39)(37,43,44,42);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,32,34,31)(30,36,38,35)(33,40,41,39)(37,43,44,42);(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)(29,30)(31,36)(32,35)(33,37)(34,38)(39,43)(40,42)(41,44);(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,33)(30,37)(31,39)(32,40)(34,41)(35,42)(36,43)(38,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 4378882);