#4867001

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 1048576
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2515 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4774197);