#364062

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1130 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)(7,16)(8,20)(9,13)(10,15)(11,18)(12,21)(14,23)(17,30)(19,25)(22,28)(24,32)(26,33)(27,36)(29,34)(31,35);(5,7)(6,11,15,26)(8,17)(9,18)(10,19)(12,27,20,24)(13,25,14,16)(21,31)(22,32)(23,33)(28,35,29,30)(34,36);(5,8,9,22,10,21,23,34)(6,12,14,28,15,20,13,29)(7,17,18,32,19,31,33,36)(11,24,16,30,26,27,25,35);(6,13,15,14)(8,21)(11,25,26,16)(12,28,20,29)(17,31)(22,34)(24,30,27,35)(32,36)

Polycyclic presentation:

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

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, 271258);