#4311721

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
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

1777 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)(10,12)(15,21)(16,23)(19,24)(20,25)(29,30,33,37)(31,35,39,34)(32,40,41,36)(38,42,44,43);(14,18)(19,24)(20,25)(26,28)(29,31,33,39)(30,34,37,35)(32,38,41,44)(36,42,40,43);(1,2,3,4)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,43)(40,42)(41,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,30)(31,34)(32,36)(33,37)(35,39)(38,42)(40,41)(43,44);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,41)(30,36,37,40)(31,38,39,44)(34,42,35,43)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4218917);