#495717

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#2193	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1038 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:

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

Polycyclic presentation:

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

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, 402913);