#368789

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
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

1773 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)(7,12)(8,20,24,13)(9,14)(10,25)(11,15)(16,26,32,36)(17,27)(18,28)(19,33)(21,34,31,29)(22,35,23,30);(2,4)(5,7)(6,12)(8,21,24,31)(9,17)(10,19)(11,18)(13,29,20,34)(14,27)(15,28)(16,23,32,22)(25,33)(26,30,36,35);(5,8,10,23,11,24,9,22)(6,13,25,35,15,20,14,30)(7,16,17,31,18,32,19,21)(12,26,27,29,28,36,33,34);(2,4)(5,9,11,10)(6,14,15,25)(7,17,18,19)(8,22,24,23)(12,27,28,33)(13,30,20,35)(16,31,32,21)(26,29,36,34)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,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	#72313084, #72313126, #72313145
maximal normal subgroup of	#72313084, #72313126, #72313145

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275985);