#494648

order	512
nilpotency class	4
isoclinism	isoclinic class of #481349
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

1300 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)(5,6,9,14,11,16,29,17)(7,18,23,35,22,33,21,26)(8,24,12,31,27,13,25,32)(10,30,28,15,19,20,36,34);(5,7,19,13,9,21,36,31,11,22,10,24,29,23,28,32)(6,12,15,33,17,27,30,26,16,25,34,18,14,8,20,35);(2,4)(5,8)(6,13)(7,20)(9,12)(10,26)(11,27)(14,32)(15,21)(16,24)(17,31)(18,28)(19,35)(22,30)(23,34)(25,29)(33,36);(5,9,11,29)(6,14,16,17)(7,21,22,23)(8,25,27,12)(10,28,19,36)(13,31,24,32)(15,20,34,30)(18,26,33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320296, #72320425, #72320546, …
maximal normal subgroup of	#72320296, #72320425, #72320546, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401844);