#4470035

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

1779 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)(6,10)(8,11)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,26,28,27)(29,30,34,32)(31,37,39,35)(33,40,42,36)(38,43,41,44);(6,10)(7,12)(14,15)(17,25)(19,21)(23,28)(29,31,34,39)(30,35,32,37)(33,41,42,38)(36,44,40,43);(5,6)(7,11)(8,12)(9,10)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31)(30,35)(32,37)(33,38)(34,39)(36,43)(40,44)(41,42);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,32,34,30)(31,35,39,37)(33,40,42,36)(38,43,41,44);(5,8,9,11)(6,12,10,7)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,36,32,40)(31,38,39,41)(35,43,37,44)

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=gh,d^2=h,e^2=g,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, 4377231);