#369050

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
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	C₂×#3153	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1844 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=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, 276246);