#5619013

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1919 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)(7,8)(9,10)(11,12)(15,21)(16,23)(19,24)(20,25)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(6,10)(8,11)(14,18)(19,24)(20,25)(26,28)(29,30,33,36)(31,34,39,37)(32,35,41,44)(38,42,40,43);(5,7)(6,11)(8,10)(9,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,31)(30,34)(32,40)(33,39)(35,43)(36,37)(38,41)(42,44);(5,8)(6,12)(7,10)(9,11)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,41)(30,35,36,44)(31,38,39,40)(34,42,37,43)

Polycyclic presentation:

2^9|b^2=fg,c^2=fgh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5526209);