#4420557

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
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

1946 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)(7,11,12,9)(14,18)(15,21)(16,23)(27,28)(29,30,34,32)(31,37,39,35)(33,36,42,40)(38,44,41,43);(13,14,17,18)(15,19,21,25)(16,20,23,26)(22,27,24,28)(29,31)(30,35)(32,37)(33,38)(34,39)(36,43)(40,44)(41,42);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,25)(22,27)(23,26)(24,28)(29,32,34,30)(31,37,39,35)(33,40,42,36)(38,44,41,43);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,17,21)(14,19,18,25)(16,24,23,22)(20,28,26,27)(29,31)(30,35)(32,37)(33,41)(34,39)(36,44)(38,42)(40,43);(13,16,17,23)(14,20,18,26)(15,22,21,24)(19,27,25,28)(29,33)(30,36)(31,38)(32,40)(34,42)(35,43)(37,44)(39,41)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=fh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4327753);