#273101

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180297);