#400695

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
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

1407 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)(7,18)(8,22)(9,16)(10,15)(11,17)(12,20)(13,21)(14,24)(19,33)(23,31)(25,30)(26,28)(27,36)(29,34)(32,35);(5,7,9,26,11,20,10,21)(6,12,15,28,17,18,16,13)(8,19,23,36,25,34,24,35)(14,27,30,33,31,32,22,29);(5,7)(6,13)(8,19)(9,26)(10,21)(11,20)(12,15)(14,29)(16,18)(17,28)(22,32)(23,36)(24,35)(25,34)(27,30)(31,33);(2,4)(5,8,9,23,11,25,10,24)(6,14,15,30,17,31,16,22)(7,19,26,36,20,34,21,35)(12,27,28,33,18,32,13,29)

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,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

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, 307891);