#300674

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
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

1035 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207870);