#272838

order	512
nilpotency class	3
isoclinism	isoclinic class of #272795
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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1241 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)(7,13,9,11)(8,16)(10,14)(12,18)(15,20,17,19)(21,22,26,30)(23,28,33,36)(24,29,25,27)(31,35,32,34);(1,2)(3,4)(5,7)(6,11)(8,15)(9,10)(12,19)(13,14)(16,20)(17,18)(22,27)(24,25)(28,34)(29,30)(31,32)(35,36);(2,4)(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,20)(14,16)(21,23,26,33)(22,28,30,36)(24,31,25,32)(27,34,29,35);(5,9,10,7)(6,13,14,11)(8,17,18,15)(12,20,16,19)(21,24)(23,31)(25,26)(27,29)(32,33)(34,35)

Polycyclic presentation:

2^9|a^2=fh,c^2=i,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=de,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, 180034);