#3070281

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

1850 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)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,27,26,28)(29,30,34,32)(31,37,39,35)(33,36,42,40)(38,44,41,43);(7,11)(9,12)(14,15)(17,25)(19,21)(23,26)(29,31,34,39)(30,35,32,37)(33,41,42,38)(36,44,40,43);(5,7)(6,9)(8,11)(10,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,27,26,28)(29,31)(30,35)(32,37)(33,38)(34,39)(36,43)(40,44)(41,42);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,24)(14,19,15,21)(17,26,25,23)(20,28,22,27)(29,32,34,30)(31,35,39,37)(33,36,42,40)(38,44,41,43);(13,17,18,25)(14,20,15,22)(16,23,24,26)(19,27,21,28)(29,33,34,42)(30,36,32,40)(31,38,39,41)(35,43,37,44)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fg,d^2=f,e^2=f,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, 2977477);