#424590

order	512
nilpotency class	3
isoclinism	isoclinic class of #417562
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1261 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331786);