#4144993

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fgh,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,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, 4052189);