#117828

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

607 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,d^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fh,d^c=di,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25024);