#409720

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

879 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)(5,6,11,15,12,7,10,16)(8,17,21,28,22,13,26,19)(9,18,24,30,25,14,23,29)(20,27,33,32,34,31,35,36);(1,2,4,3)(5,7,12,6)(8,19,22,28)(9,18,25,14)(10,15,11,16)(13,21,17,26)(20,32,34,36)(23,29,24,30)(27,33,31,35);(5,8)(6,13)(7,17)(9,20)(10,21)(11,26)(12,22)(14,27)(15,28)(16,19)(18,31)(23,33)(24,35)(25,34)(29,36)(30,32);(1,3,4,2)(5,9,10,23,12,25,11,24)(6,14,16,30,7,18,15,29)(8,20,21,33,22,34,26,35)(13,27,19,32,17,31,28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316916);