#400696

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	#23438	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1397 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)(7,17)(8,22)(9,15)(10,16)(11,19)(12,20)(13,24)(14,26)(18,29)(21,28)(23,36)(25,31)(27,34)(30,35)(32,33);(5,7,9,21,10,19,26,20)(6,11,14,28,16,17,15,12)(8,23,24,34,25,18,33,35)(13,30,22,29,32,27,31,36);(5,7)(6,12)(8,18)(9,21)(10,19)(11,14)(13,29)(15,17)(16,28)(20,26)(22,27)(23,25)(24,35)(30,31)(32,36)(33,34);(2,4)(5,8,9,24,10,25,26,33)(6,13,14,22,16,32,15,31)(7,18,21,35,19,23,20,34)(11,27,28,36,17,30,12,29)

Polycyclic presentation:

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

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, 307892);