#494455

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1392 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi

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, 401651);