#418479

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1657 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 325675);