#364180

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1425 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:

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

Polycyclic presentation:

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

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, 271376);