#4342127

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fh,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,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, 4249323);