#333585

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28397	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

905 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240781);