#340309

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

641 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)(7,14,16,18)(8,10,17,12)(11,19,20,15)(21,22,26,30)(23,32,24,28)(25,31,27,35)(29,33,36,34);(1,2)(3,4)(6,10)(7,15)(8,17)(12,13)(14,18)(16,19)(21,23)(22,28)(24,26)(25,34)(27,33)(29,35)(30,32)(31,36);(2,4)(7,16)(11,20)(14,18)(15,19)(21,24,26,23)(22,29,30,36)(25,34,27,33)(28,31,32,35);(5,7,8,15,9,16,17,19)(6,11,12,14,13,20,10,18)(21,25)(22,30)(23,33)(24,34)(26,27)(29,36)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=ei,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=de,e^b=eh,d^c=dh,f^c=fi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247505);