#4435220

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

2239 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)(5,6,9,11)(7,12,8,10)(14,18)(16,19)(22,26)(23,29)(28,31)(30,35);(5,6)(7,12)(8,10)(9,11)(13,14,17,18)(15,16,20,19)(21,22)(23,27)(24,28)(25,26)(29,33)(30,34)(31,32)(35,36);(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,18)(17,20)(21,23,25,29)(22,27,26,33)(24,30,32,35)(28,34,31,36);(1,2,3,4)(5,8,9,7)(6,10,11,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,27)(24,31)(25,26)(28,32)(29,33)(30,36)(34,35);(5,6)(7,10)(8,12)(9,11)(13,16)(14,20)(15,18)(17,19)(21,24,25,32)(22,28,26,31)(23,30,29,35)(27,34,33,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=g,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, 4342416);