#247441

order	512
nilpotency class	3
isoclinism	isoclinic class of #247367
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1107 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)(7,14)(8,10)(9,13)(11,15)(12,17)(16,20)(18,19)(21,22,28,34)(23,31,35,30)(24,29,25,36)(26,33,27,32);(1,2)(3,4)(6,10)(8,17)(11,14)(12,13)(15,18)(19,20)(21,23,28,35)(22,29,34,36)(24,26,25,27)(30,33,31,32);(2,4)(21,24,28,25)(22,30,34,31)(23,26,35,27)(29,33,36,32);(5,7,9,16)(6,11,13,19)(8,18,17,15)(10,14,12,20)(21,25,28,24)(22,31,34,30)(23,26,35,27)(29,33,36,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,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^c=fi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154637);