#116096

order	512
nilpotency class	3
isoclinism	isoclinic class of #10004
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

736 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)(5,6)(7,11)(8,10)(9,16)(12,15)(13,19)(14,18)(17,20)(21,22)(23,29)(24,32)(25,28)(26,30)(27,33)(31,36)(34,35);(6,10)(7,13,14,17)(11,19,18,20)(12,16)(24,33)(26,35)(27,32)(30,34);(5,7)(6,11)(8,14)(9,17)(10,18)(12,19)(13,15)(16,20)(21,23,25,31)(22,26,28,35)(24,34,33,30)(27,36,32,29)

Polycyclic presentation:

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

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, 23292);