#340331

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

673 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)(21,22,25,29)(23,26,31,35)(24,27,32,28)(30,33,36,34);(1,2)(3,4)(6,10)(7,15)(8,17)(12,13)(14,18)(16,19)(21,23,25,31)(22,26,29,35)(24,30,32,36)(27,33,28,34);(2,4)(7,16)(11,20)(14,18)(15,19)(22,27)(24,32)(26,33)(28,29)(30,36)(34,35);(5,7,8,15,9,16,17,19)(6,11,12,14,13,20,10,18)(22,27)(24,32)(26,33)(28,29)(30,36)(34,35)

Polycyclic presentation:

2^9|a^2=hi,b^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=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, 247527);