#267752

order	512
nilpotency class	3
isoclinism	isoclinic class of #267659
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

1101 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)(8,16)(9,19,20,13)(10,14)(12,18)(21,22,26,34,25,29,23,27)(24,31,30,28,33,35,32,36);(1,2)(3,4)(5,7)(6,11)(8,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23,25,26)(22,27,29,34)(24,30,33,32)(28,35,36,31);(2,4)(5,8)(6,12)(7,13)(9,15)(10,18)(11,20)(14,16)(17,19)(21,24)(22,28)(23,30)(25,33)(26,32)(27,35)(29,36)(31,34);(5,9,10,20)(6,13,14,19)(7,16,17,12)(8,15,18,11)(21,23,25,26)(22,27,29,34)(24,30,33,32)(28,35,36,31)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174948);