#5478824

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

1712 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)(13,14,18,19)(15,20,22,16)(17,21,25,27)(23,28,26,24)(29,30)(31,35)(32,34)(33,40)(36,39)(37,42)(38,44)(41,43);(6,9)(10,12)(14,19)(15,22)(17,25)(24,28)(29,31,34,39)(30,35,32,36)(33,41,42,38)(37,44,40,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,19)(17,26)(18,22)(21,24)(23,25)(27,28)(29,31)(30,36)(32,35)(33,38)(34,39)(37,44)(40,43)(41,42);(1,2)(3,4)(13,16,18,20)(14,15,19,22)(17,24,25,28)(21,23,27,26)(29,32,34,30)(31,36,39,35)(33,40,42,37)(38,44,41,43);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,21,19,27)(15,23,22,26)(16,24,20,28)(29,33)(30,37)(31,38)(32,40)(34,42)(35,43)(36,44)(39,41)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=fg,e^2=gh,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, 5386020);