#298533

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
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×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1042 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)(7,16)(8,17)(9,15)(10,14)(11,13)(12,18)(19,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(1,2,4,3)(5,7)(6,12)(8,13)(9,17)(10,18)(11,20)(14,16)(15,19)(21,23)(22,26)(24,31)(25,30)(27,29)(28,34)(32,33)(35,36);(5,8,11,16,10,19,9,12)(6,7,15,17,14,18,13,20)(23,30)(25,33)(26,34)(28,36)(29,35)(31,32);(1,3,4,2)(5,9,10,11)(6,13,14,15)(7,17,18,20)(8,16,19,12)(22,27)(26,29)(28,36)(34,35)

Polycyclic presentation:

2^9|b^2=ehi,c^2=f,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

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, 205729);