#306735

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,8)(5,7)(9,10)(11,18)(12,16)(13,15)(14,19)(17,22)(20,24)(21,23)(25,26,30,34)(27,31,37,40)(28,32,29,33)(35,38,36,39);(1,3,5,4)(2,6,7,8)(9,11,13,20)(10,14,15,23)(12,21,22,19)(16,18,17,24)(25,27,30,37)(26,31,34,40)(28,36,29,35)(32,39,33,38);(1,4,5,3)(2,6,7,8)(10,15)(14,23)(16,17)(18,24)(27,35)(28,29)(31,38)(32,33)(36,37)(39,40);(10,16)(12,22)(14,18)(15,17)(19,21)(23,24)(25,28,30,29)(26,32,34,33)(27,36,37,35)(31,39,40,38)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fh,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gi,d^c=di,g^c=gi,e^d=eh

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, 213931);