#494153

order	512
nilpotency class	4
isoclinism	isoclinic class of #483399
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

734 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224826, #72224931, #72224942, …
maximal normal subgroup of	#72224826, #72224931, #72224942, …
maximal quotient of	#72224817, #72310108, #72319118, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401349);