#359157

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1382 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)(5,6,10,15,11,16,9,23)(7,12,18,13,19,27,24,26)(8,14,21,29,22,30,20,33)(17,25,31,28,32,36,34,35);(1,2)(3,4)(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,15)(14,25)(16,27)(20,34)(21,31)(22,32)(23,26)(28,29)(30,36)(33,35);(1,3,4,2)(5,7)(6,13)(8,17)(9,24)(10,18)(11,19)(12,23)(14,28)(15,27)(16,26)(20,34)(21,31)(22,32)(25,33)(29,36)(30,35);(5,8,10,21,11,22,9,20)(6,14,15,29,16,30,23,33)(7,17,18,31,19,32,24,34)(12,25,13,28,27,36,26,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266353);