#4419566

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

1817 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)(5,6,9,12)(7,10,8,11)(13,14,18,15)(16,21,24,19)(17,22,26,20)(23,27,25,28)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,7,9,8)(6,10,12,11)(14,15)(19,21)(20,22)(27,28)(31,38)(35,37)(39,44)(42,43);(5,6)(7,10)(8,11)(9,12)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27)(29,31)(30,35)(32,37)(33,39)(34,38)(36,42)(40,43)(41,44);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,32,34,30)(31,37,38,35)(33,40,41,36)(39,43,44,42);(5,8,9,7)(6,11,12,10)(13,17,18,26)(14,20,15,22)(16,23,24,25)(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=fg,b^2=fgh,c^2=f,d^2=h,e^2=fg,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, 4326762);