#4385571

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 2097152
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

2041 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)(6,8)(7,11)(14,18)(16,19)(21,22,26,27)(23,31,32,28)(24,29,33,34)(25,30,35,36);(5,6,9,8)(7,12,11,10)(13,14)(15,16)(17,18)(19,20)(22,27)(24,33)(28,31)(30,36);(1,2,3,4)(6,8)(10,12)(14,18)(16,19)(21,23,26,32)(22,28,27,31)(24,30,33,36)(25,34,35,29);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20)(21,24)(22,29)(23,30)(25,28)(26,33)(27,34)(31,35)(32,36);(5,8,9,6)(7,12,11,10)(13,16,17,19)(14,15,18,20)(21,25)(22,30)(23,29)(24,28)(26,35)(27,36)(31,33)(32,34)

Polycyclic presentation:

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

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, 4292767);