#244925

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3305	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1168 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152121);