#4342114

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1745 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)(13,14)(15,21)(16,24)(17,18)(19,23)(20,26)(22,27)(25,28)(30,32)(33,41)(35,37)(38,44);(14,18)(19,21)(20,24)(27,28)(29,30,34,32)(31,37,39,35)(33,40,41,36)(38,42,44,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,26)(24,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,22)(17,26)(18,24)(19,27)(21,28)(23,25)(29,32,34,30)(31,35,39,37)(33,40,41,36)(38,42,44,43);(5,7,8,11)(6,10,9,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,33,34,41)(30,36,32,40)(31,38,39,44)(35,42,37,43)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=gi,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, 4249310);