#5474687

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

1825 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)(5,6,8,10)(7,9,11,12)(13,14,18,19)(15,20,22,16)(17,21,25,27)(23,28,26,24)(29,30,34,37)(31,35,32,39)(33,36,41,44)(38,42,40,43);(7,11)(9,12)(14,19)(15,22)(17,25)(24,28)(29,31,34,32)(30,35,37,39)(33,40,41,38)(36,43,44,42);(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,19)(17,26)(18,22)(21,24)(23,25)(27,28)(29,32,34,31)(30,35,37,39)(33,40,41,38)(36,42,44,43);(1,2)(3,4)(13,16,18,20)(14,15,19,22)(17,24,25,28)(21,23,27,26)(29,30)(31,35)(32,39)(33,36)(34,37)(38,42)(40,43)(41,44);(5,6)(7,9)(8,10)(11,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,27)(20,28)(22,26)(29,33,34,41)(30,36,37,44)(31,38,32,40)(35,42,39,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=f,e^2=g,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, 5381883);