#272729

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1048 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)(7,13,15,11)(8,10)(9,16,17,12)(14,18)(19,20)(21,22)(23,31)(24,33)(25,35)(26,28)(27,32)(29,34)(30,36);(1,2)(3,4)(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,23,26,24)(22,27,28,29)(25,34,36,32)(30,33,35,31);(2,4)(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,19,20,18)(22,28)(27,29)(30,35)(31,33);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,24,26,23)(22,29,28,27)(25,32,36,34)(30,31,35,33)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=hi,d^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, 179925);