#375879

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1686 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283075);