#368730

order	512
nilpotency class	3
isoclinism	isoclinic class of #196299
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1799 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312522, #72312585, #72312596
maximal normal subgroup of	#72312522, #72312585, #72312596

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275926);