#4435144

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

1753 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)(7,11)(10,12)(13,14)(15,18)(16,21)(17,22)(19,24)(20,26)(23,27)(25,28)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(5,6,9,8)(7,10,11,12)(14,15)(19,21)(20,22)(27,28)(29,30,33,36)(31,34,39,37)(32,35,41,40)(38,42,44,43);(5,7)(6,10)(8,12)(9,11)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27)(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)(14,19)(15,21)(17,25)(18,24)(20,28)(22,27)(23,26)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(13,17)(14,20)(15,22)(16,23)(18,26)(19,27)(21,28)(24,25)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=f,d^2=gi,e^2=fh,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, 4342340);