#4435135

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=f,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, 4342331);