#273009

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
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	C₂×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1033 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)(7,15,17,11)(8,12)(9,19,20,13)(10,14)(16,18)(21,22)(23,32)(24,30)(25,35)(26,29)(27,31)(28,33)(34,36);(1,2)(3,4)(5,7,10,17)(6,11,14,15)(8,13,18,19)(9,16,20,12)(21,23,27,25)(22,28,31,36)(24,33,26,34)(29,32,30,35);(2,4)(5,8,10,18)(6,12,14,16)(7,13,17,19)(9,15,20,11)(21,24,27,26)(22,29,31,30)(23,33,25,34)(28,32,36,35);(5,9)(6,13)(7,16)(8,15)(10,20)(11,18)(12,17)(14,19)(21,25,27,23)(22,28,31,36)(24,34,26,33)(29,32,30,35)

Polycyclic presentation:

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

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, 180205);