#1739009

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
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

1764 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)(5,6,9,8)(7,10,12,11)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28)(29,30,34,35)(31,36,40,43)(32,33,37,42)(38,44,41,39);(5,6,9,8)(7,11,12,10)(14,18)(16,25)(19,21)(22,28)(30,35)(33,42)(36,43)(39,44);(5,7,9,12)(6,10,8,11)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,31)(30,36)(32,41)(33,39)(34,40)(35,43)(37,38)(42,44);(1,2)(3,4)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,32,34,37)(30,33,35,42)(31,38,40,41)(36,44,43,39);(5,8,9,6)(7,11,12,10)(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)(29,33)(30,37)(31,39)(32,35)(34,42)(36,38)(40,44)(41,43)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=gh,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 1646205);