#387803

order	512
nilpotency class	3
isoclinism	isoclinic class of #32696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

623 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)(8,21)(9,24)(10,27)(11,15)(12,19)(13,23)(14,26)(17,29)(18,36)(20,33)(22,32)(25,31)(28,35)(30,34);(5,7,11,20)(6,12)(8,17)(9,25)(10,19,26,34)(13,31,21,28)(14,16)(15,30)(18,23)(22,35)(24,29,32,36)(27,33);(5,8,10,22,11,23,26,9)(6,13,14,32,15,21,27,24)(7,17,19,35,20,18,34,25)(12,28,16,29,30,31,33,36);(5,9,26,23,11,22,10,8)(6,13,14,32,15,21,27,24)(7,18,19,25,20,17,34,35)(12,29,33,28,30,36,16,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294999);