#340313

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

661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,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, 247509);