#4986430

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2141 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)(8,11,12,10)(13,14,18,19)(15,16,17,20)(21,22,25,26)(23,29,31,27)(24,32,33,28)(30,34,36,35);(5,6,9,7)(8,10,12,11)(13,15,18,17)(14,16,19,20)(22,26)(27,29)(28,32)(34,35);(6,7)(10,11)(14,19)(16,20)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,20)(14,15,19,17)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,8,9,12)(6,10,7,11)(13,17,18,15)(14,20,19,16)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,d^2=fhi,e^2=fg,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, 4893626);