#4435214

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
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

1682 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)(5,6,9,8)(7,11,12,10)(13,14)(15,20)(16,25)(17,24)(18,19)(21,27)(22,28)(23,26)(31,37)(33,41)(35,38)(36,40);(5,6,9,8)(7,10,12,11)(14,19)(15,23)(21,25)(22,24)(29,30,34,32)(31,38,37,35)(33,36,41,40)(39,44,43,42);(5,7,9,12)(6,10,8,11)(13,15,18,23)(14,20,19,26)(16,24,27,22)(17,21,28,25)(29,31,34,37)(30,35,32,38)(33,39,41,43)(36,42,40,44);(1,2,3,4)(5,8,9,6)(7,11,12,10)(13,16,18,27)(14,21,19,25)(15,24,23,22)(17,20,28,26)(29,32,34,30)(31,38,37,35)(33,40,41,36)(39,44,43,42);(13,17,18,28)(14,22,19,24)(15,21,23,25)(16,26,27,20)(29,33,34,41)(30,36,32,40)(31,39,37,43)(35,42,38,44)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=fg,d^2=gi,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, 4342410);