#272771

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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

1293 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179967);