#5632224

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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

1818 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)(10,11)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30)(31,34)(32,40)(33,37)(35,39)(36,42)(38,44)(41,43);(5,6)(7,10)(8,9)(11,12)(15,21)(19,22)(23,27)(26,28)(29,31,33,39)(30,34,37,35)(32,41,42,38)(36,44,40,43);(6,9)(10,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,31)(30,35)(32,38)(33,39)(34,37)(36,44)(40,43)(41,42);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,30)(31,34)(32,36)(33,37)(35,39)(38,43)(40,42)(41,44);(5,7)(6,10)(8,12)(9,11)(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,32)(30,36)(31,38)(33,42)(34,43)(35,44)(37,40)(39,41)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5539420);