#371019

order	512
nilpotency class	3
isoclinism	isoclinic class of #23990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1285 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:

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

Polycyclic presentation:

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

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, 278215);