#376487

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1091 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)(7,18,20,12)(8,21)(9,25,26,14)(10,17)(11,16)(13,24)(15,27)(19,31,36,32)(22,34,35,29)(23,30)(28,33);(1,2)(3,4)(5,7,11,19,10,20,27,36)(6,12,15,31,17,18,16,32)(8,22,24,26,23,35,28,9)(13,25,21,29,33,14,30,34);(2,4)(5,8,10,23)(6,13,17,33)(7,9,20,26)(11,28,27,24)(12,29,18,34)(14,31,25,32)(15,30,16,21)(19,35,36,22);(5,9,10,26)(6,14)(7,8,20,23)(11,22,27,35)(12,30)(13,31)(15,34)(16,29)(17,25)(18,21)(19,24,36,28)(32,33)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,f^c=fi,g^c=gi,e^d=ei,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, 283683);