#353555

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1806 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260751);