#146438

order	512
nilpotency class	4
isoclinism	isoclinic class of #12833
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1807
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250554, #72250580, #72250583, …
maximal normal subgroup of	#72250554, #72250580, #72250583, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53634);