#246715

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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	#59049	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1277 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:

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

Polycyclic presentation:

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

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, 153911);