#146535

order	512
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1823
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

994 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)(7,16,18,12)(8,20)(9,24)(10,14)(11,17,27,34)(13,22)(15,26)(19,30,25,31)(21,28,35,29)(23,33)(32,36);(5,7)(6,11,14,27)(8,21)(9,25)(10,18)(12,20,16,33)(13,31,32,30)(15,29,24,28)(17,22)(19,26)(23,35)(34,36);(2,4)(5,7)(6,12)(8,17)(9,19)(10,18)(11,20)(13,29)(14,16)(15,30)(21,22)(23,34)(24,31)(25,26)(27,33)(28,32)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231520, #72231524, #72231555, …
maximal normal subgroup of	#72231520, #72231524, #72231555, …
maximal quotient of	#72265305, #72280528, C₂×#146535

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53731);