#117206

order	512
nilpotency class	3
isoclinism	isoclinic class of #10104
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

686 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)(5,6,10,13)(7,14,12,9)(8,11,17,19)(15,16,20,18)(21,22)(23,31)(24,29)(25,28)(26,30)(27,33)(32,35)(34,36);(5,7,9,6,10,12,14,13)(8,15,16,19,17,20,18,11)(21,23,24,33,26,32,25,34)(22,27,28,31,30,36,29,35);(5,8,10,17)(6,11,13,19)(7,15,12,20)(9,18,14,16)(22,28)(23,32)(24,25)(27,35)(29,30)(31,36)

Polycyclic presentation:

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

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, 24402);