#4939467

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
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

2058 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)(5,6,9,7)(8,10,12,11)(13,14,17,18)(15,16,19,20)(21,22,27,25)(23,29,24,28)(26,33,34,30)(31,35,32,36);(13,15)(14,16)(17,19)(18,20)(21,23,27,24)(22,28,25,29)(26,32,34,31)(30,36,33,35);(6,7)(10,11)(14,18)(15,19)(21,24,27,23)(22,29,25,28)(26,32,34,31)(30,36,33,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,19)(18,20)(21,25,27,22)(23,29,24,28)(26,33,34,30)(31,36,32,35);(5,8,9,12)(6,10,7,11)(13,16)(14,15)(17,20)(18,19)(21,26)(22,30)(23,31)(24,32)(25,33)(27,34)(28,35)(29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4846663);