#246910

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#46134	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154106);