#376170

order	512
nilpotency class	3
isoclinism	isoclinic class of #23990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1602 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)(7,16,20,27)(8,21)(9,12)(10,15)(11,25)(13,19,26,17)(14,22)(18,31,33,36)(23,30)(24,29)(28,34,35,32);(1,2,4,3)(6,12)(7,17,20,19)(13,27,26,16)(14,21)(15,25)(18,32,33,34)(28,36,35,31)(29,30);(5,7)(6,13)(8,18)(9,20)(10,17)(11,19)(12,26)(14,28)(15,27)(16,25)(21,35)(22,33)(23,34)(24,32)(29,31)(30,36);(1,3,4,2)(5,8,9,22)(6,14,12,21)(7,18,20,33)(10,24,11,23)(13,28,26,35)(15,30,25,29)(16,31,27,36)(17,32,19,34)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=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, 283366);