#388702

order	512
nilpotency class	3
isoclinism	isoclinic class of #31282
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

823 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=h,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,e^d=ei,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, 295898);