#5474615

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

1852 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)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(7,11)(9,12)(14,19)(15,22)(17,25)(24,28)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,19)(17,26)(18,22)(21,24)(23,25)(27,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(1,2)(3,4)(13,16,18,20)(14,15,19,22)(17,24,25,28)(21,23,27,26)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,6)(7,9)(8,10)(11,12)(13,17,18,25)(14,21,19,27)(15,23,22,26)(16,24,20,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=f,e^2=fg,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, 5381811);