#115847

order	512
nilpotency class	3
isoclinism	isoclinic class of #100003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23043);