#117871

order	512
nilpotency class	3
isoclinism	isoclinic class of #10128
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

636 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)(5,6,11,13)(7,9,15,10)(8,16,19,14)(12,17,20,18)(21,22)(23,25)(24,32)(26,29)(27,30)(28,34)(31,35)(33,36);(5,7,9,13,11,15,10,6)(8,12,17,14,19,20,18,16)(21,23,26,30,27,29,25,22)(24,31,34,28,35,32,33,36);(5,8,11,19)(6,12,13,20)(7,14,15,16)(9,17,10,18)(21,24,27,35)(22,28,30,36)(23,31,29,32)(25,34,26,33)

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=h,d^2=h,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,f^b=fh,d^c=di,e^c=eh,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, 25067);