#494475

order	512
nilpotency class	4
isoclinism	isoclinic class of #481319
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3061	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285780, #72320284, #72320546, …
maximal normal subgroup of	#72285780, #72320284, #72320546, …
maximal quotient of	#72286006, #72286976, #72287176

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401671);