#4157569

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1730 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)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,27,26,28)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,7,9,12)(6,10,8,11)(14,18)(16,25)(19,21)(22,26)(31,38)(33,41)(35,37)(36,40);(5,7)(6,10)(8,11)(9,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(29,31,34,38)(30,35,32,37)(33,39,41,44)(36,42,40,43);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,15,17,23)(14,19,18,21)(16,26,25,22)(20,28,24,27)(29,32,34,30)(31,35,38,37)(33,40,41,36)(39,42,44,43);(5,8,9,6)(7,11,12,10)(13,16,17,25)(14,20,18,24)(15,22,23,26)(19,27,21,28)(29,33,34,41)(30,36,32,40)(31,39,38,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=h,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, 4064765);