#4435224

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1837 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)(6,9)(10,11)(13,14,17,16)(15,19,20,18)(21,22,25,26)(23,29,31,27)(24,32,34,28)(30,35,33,36);(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,18,20,19)(22,26)(27,29)(28,32)(35,36);(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,18,16,19)(21,22)(23,27)(24,28)(25,26)(29,31)(30,35)(32,34)(33,36);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,16,17,14)(15,19,20,18)(21,23)(22,27)(24,33)(25,31)(26,29)(28,36)(30,34)(32,35);(5,7,8,12)(6,10,9,11)(21,24)(22,28)(23,30)(25,34)(26,32)(27,35)(29,36)(31,33)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=fg,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4342420);