#364467

order	512
nilpotency class	3
isoclinism	isoclinic class of #196134
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2460 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)(8,20,23,13)(9,14)(10,24)(11,15)(16,26,32,36)(17,27)(18,28)(19,33)(21,35,34,29)(22,31,25,30);(2,4)(5,7)(6,12)(8,21,23,34)(9,17)(10,19)(11,18)(13,29,20,35)(14,27)(15,28)(16,22,32,25)(24,33)(26,31,36,30);(5,8)(6,13)(7,16)(9,22)(10,25)(11,23)(12,26)(14,30)(15,20)(17,21)(18,32)(19,34)(24,31)(27,35)(28,36)(29,33);(2,4)(5,9,11,10)(6,14,15,24)(7,17,18,19)(8,22,23,25)(12,27,28,33)(13,30,20,31)(16,21,32,34)(26,35,36,29)

Polycyclic presentation:

2^9|a^2=h,b^2=h,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,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313084, #72313130, #72313145
maximal normal subgroup of	#72313084, #72313130, #72313145

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271663);