#116238

order	512
nilpotency class	3
isoclinism	isoclinic class of #10018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

704 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23434);