#298318

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

980 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=f,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,g^c=gh

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, 205514);