#383239

order	512
nilpotency class	3
isoclinism	isoclinic class of #30496
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1869 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)(5,6,11,15,12,8,10,16)(7,17,20,22,21,32,19,13)(9,23,26,31,27,14,25,30)(18,29,34,36,35,33,24,28);(5,7,10,19,12,21,11,20)(6,13,15,17,8,22,16,32)(9,24,25,35,27,34,26,18)(14,29,30,36,23,33,31,28);(2,4)(5,8)(6,12)(7,17)(9,23)(10,16)(11,15)(13,20)(14,27)(18,33)(19,22)(21,32)(24,28)(25,31)(26,30)(29,35)(34,36);(2,4)(5,9,12,27)(6,14,8,23)(7,18,21,35)(10,26,11,25)(13,28,22,36)(15,31,16,30)(17,33,32,29)(19,34,20,24)

Polycyclic presentation:

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

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, 290435);