#4314978

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

2130 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)(8,12)(13,14,18,17)(15,20,16,19)(21,22)(23,28)(24,25)(26,27)(29,34)(30,36)(31,33)(32,35);(5,6,9,7)(8,11,12,10)(22,27)(25,29)(28,35)(31,36);(6,7)(10,11)(13,15,18,16)(14,19,17,20)(21,23)(22,28)(24,33)(25,31)(26,32)(27,35)(29,36)(30,34);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,20,17,19)(21,24)(22,29)(23,30)(25,27)(26,34)(28,31)(32,33)(35,36);(5,8,9,12)(6,10,7,11)(13,17,18,14)(15,20,16,19)(21,25,26,29)(22,24,27,34)(23,31,32,36)(28,33,35,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4222174);