#374384

order	512
nilpotency class	3
isoclinism	isoclinic class of #196110
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1680 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=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, 281580);