#146551

order	512
nilpotency class	4
isoclinism	isoclinic class of #12933
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1853
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

990 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)(7,17,20,33)(8,21)(9,15)(10,27)(11,16)(12,18,30,35)(13,19,32,26)(14,24)(22,31,36,29)(23,28)(25,34);(5,7)(6,12,16,30)(8,22)(9,19)(10,26)(11,20)(13,28,32,14)(15,29,27,31)(17,34,33,21)(18,24)(23,35)(25,36);(2,4)(5,7)(6,13)(8,18)(9,26)(10,19)(11,20)(12,28)(14,30)(15,33)(16,32)(17,27)(21,29)(22,24)(23,36)(25,35)(31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231513, #72231535, #72231555, …
maximal normal subgroup of	#72231513, #72231535, #72231555, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53747);