#4464096

order	512
nilpotency class	2
isoclinism	isoclinic class of #4462490
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1932 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)(15,18)(16,21)(17,22)(19,24)(20,25)(23,26)(27,28)(30,32)(31,37)(36,40)(38,43);(7,11)(10,12)(14,15)(19,21)(20,22)(26,27)(29,30,34,32)(31,35,37,39)(33,36,41,40)(38,42,43,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31)(30,35)(32,39)(33,38)(34,37)(36,42)(40,44)(41,43);(1,2,3,4)(5,7,8,11)(6,10,9,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,32,34,30)(31,35,37,39)(33,40,41,36)(38,42,43,44);(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(29,33,34,41)(30,36,32,40)(31,38,37,43)(35,42,39,44)

Polycyclic presentation:

2^9|b^2=fh,c^2=h,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,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, 4371292);