#340419

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

653 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=dei,e^b=eh,d^c=dh,f^c=fi,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247615);