#3498018

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

2294 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3405214);