#3078330

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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

1706 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)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,27,26,28)(29,30,34,38)(31,36,32,35)(33,37,42,41)(39,44,40,43);(6,9)(10,12)(14,15)(17,25)(19,21)(23,26)(29,31,34,32)(30,35,38,36)(33,40,42,39)(37,44,41,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,27,26,28)(29,32,34,31)(30,36,38,35)(33,40,42,39)(37,44,41,43);(1,2)(3,4)(13,16,18,24)(14,19,15,21)(17,26,25,23)(20,28,22,27)(29,30)(31,36)(32,35)(33,41)(34,38)(37,42)(39,43)(40,44);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,20,15,22)(16,23,24,26)(19,27,21,28)(29,33,34,42)(30,37,38,41)(31,39,32,40)(35,43,36,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2985526);