#368732

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

1780 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)(5,6,10,15,11,16,9,14)(7,12,19,28,20,29,18,27)(8,21,23,31,25,32,24,13)(17,33,34,26,35,30,22,36);(2,4)(5,7,10,19,11,20,9,18)(6,12,15,28,16,29,14,27)(8,22,23,17,25,34,24,35)(13,30,21,36,31,33,32,26);(5,8)(6,13,16,31)(7,17,20,35)(9,23)(10,24)(11,25)(12,26)(14,21,15,32)(18,34,19,22)(27,30)(28,33)(29,36);(2,4)(5,9,11,10)(6,14,16,15)(7,18,20,19)(8,23,25,24)(12,27,29,28)(13,21,31,32)(17,34,35,22)(26,30,36,33)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=i,f^2=i,g^2=i,h^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	#72313084, #72313132, #72313151, …
maximal normal subgroup of	#72313084, #72313132, #72313151, …
maximal quotient of	#72316874, #72316996, #72317024

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275928);