#246782

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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

1122 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:

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

Polycyclic presentation:

2^9|a^2=h,b^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,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153978);