#4980311

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

2098 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)(7,9)(8,10)(11,12)(15,17)(19,20)(21,22,25,26)(23,27,30,32)(24,28,31,33)(29,34,35,36);(5,7)(6,9)(8,12)(10,11)(13,14,18,16)(15,20,17,19)(22,26)(24,31)(27,32)(29,35);(2,3)(6,8)(7,11)(13,15,18,17)(14,19,16,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(1,3,4,2)(5,8,10,6)(7,12,11,9)(13,16,18,14)(15,19,17,20)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(5,9,10,12)(6,11,8,7)(13,17,18,15)(14,19,16,20)(21,24)(22,28)(23,29)(25,31)(26,33)(27,34)(30,35)(32,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=fgi,d^2=g,e^2=fh,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, 4887507);