#4456676

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 2097152
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

1860 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df

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, 4363872);