#2777994

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
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

1646 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=g,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,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, 2685190);