#3468944

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

1762 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)(7,11)(10,12)(14,15)(16,23)(17,25)(27,28)(29,30,33,37)(31,35,39,34)(32,36,42,41)(38,44,40,43);(6,9)(10,12)(13,14,18,15)(16,19,23,21)(17,20,25,22)(24,27,26,28)(29,31,33,39)(30,34,37,35)(32,38,42,40)(36,43,41,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,27,26,28)(29,31)(30,35)(32,40)(33,39)(34,37)(36,43)(38,42)(41,44);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,19)(15,21)(17,26)(18,23)(20,28)(22,27)(24,25)(29,30)(31,35)(32,41)(33,37)(34,39)(36,42)(38,43)(40,44);(13,17,18,25)(14,20,15,22)(16,24,23,26)(19,27,21,28)(29,32)(30,36)(31,38)(33,42)(34,43)(35,44)(37,41)(39,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=g,d^2=ghi,e^2=g,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, 3376140);