#4435107

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

1932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=g,d^2=i,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, 4342303);