#5480990

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

1669 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)(7,11)(10,12)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(29,30,34,37)(31,35,39,32)(33,36,42,44)(38,43,41,40);(6,9)(10,12)(15,22)(17,25)(19,21)(20,24)(29,31,34,39)(30,32,37,35)(33,41,42,38)(36,43,44,40);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28)(29,31)(30,35)(32,37)(33,41)(34,39)(36,40)(38,42)(43,44);(1,2)(3,4)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,32,34,35)(30,31,37,39)(33,40,42,43)(36,38,44,41);(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33)(30,36)(31,38)(32,40)(34,42)(35,43)(37,44)(39,41)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fgh,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=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, 5388186);