#368475

order	512
nilpotency class	3
isoclinism	isoclinic class of #24420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1343 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=g,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275671);