#107140

order	512
nilpotency class	3
isoclinism	isoclinic class of #8453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14336);