#392828

order	512
nilpotency class	3
isoclinism	isoclinic class of #32947
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1535 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280919, #72281182, #72281232, …
maximal normal subgroup of	#72280919, #72281182, #72281232, …
maximal quotient of	#72313650, #72343141

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300024);