#355850

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1169 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263046);