#4157570

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1746 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)(5,6,9,8)(7,10,12,11)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(29,30)(31,32)(33,41)(34,35)(36,37)(38,42)(39,44)(40,43);(5,7,9,12)(6,10,8,11)(15,22)(17,25)(19,21)(20,24)(30,35)(31,37)(33,42)(40,44);(5,7)(6,10)(8,11)(9,12)(13,15,18,22)(14,19,16,21)(17,23,25,28)(20,26,24,27)(29,31)(30,36)(32,35)(33,43)(34,37)(38,40)(39,42)(41,44);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,32,34,36)(30,37,35,31)(33,44,42,40)(38,39,41,43);(5,8,9,6)(7,11,12,10)(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)(29,33,34,42)(30,38,35,41)(31,39,37,43)(32,40,36,44)

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=fg,d^2=h,e^2=h,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, 4064766);