#368786

order	512
nilpotency class	3
isoclinism	isoclinic class of #196350
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1750 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)(3,4)(5,6,9,14,10,13,21,27)(7,11,17,24,16,23,31,35)(8,12,19,26,18,25,20,28)(15,22,30,33,29,34,32,36);(2,4)(5,7,9,17,10,16,21,31)(6,11,14,24,13,23,27,35)(8,15,19,30,18,29,20,32)(12,22,26,33,25,34,28,36);(5,8)(6,12,13,25)(7,15,16,29)(9,20)(10,18)(11,22)(14,28,27,26)(17,32,31,30)(19,21)(23,34)(24,36)(33,35);(2,4)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,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, 275982);