#3353802

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1697 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)(13,14,18,16)(15,19,21,22)(17,20,26,24)(23,27,25,28)(29,30,34,32)(31,37,39,35)(33,40,42,36)(38,43,41,44);(6,9)(10,12)(14,16)(15,21)(20,24)(23,25)(29,31,34,39)(30,35,32,37)(33,38,42,41)(36,43,40,44);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,22)(17,25)(18,21)(20,28)(23,26)(24,27)(29,31)(30,35)(32,37)(33,41)(34,39)(36,44)(38,42)(40,43);(1,2)(3,4)(13,16,18,14)(15,22,21,19)(17,20,26,24)(23,27,25,28)(29,32,34,30)(31,37,39,35)(33,36,42,40)(38,43,41,44);(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28)(29,33,34,42)(30,36,32,40)(31,38,39,41)(35,43,37,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3260998);