#340421

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, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^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=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, 247617);