#467935

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

736 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)(5,6,11,16)(7,13,8,20)(9,14)(10,27,17,15)(12,22,29,19)(18,34)(21,30)(23,36)(24,33)(25,32)(26,31)(28,35);(5,7,17,22,11,8,10,19)(6,12,27,20,16,29,15,13)(9,18,24,35,25,21,26,23)(14,28,31,34,32,36,33,30);(1,2,4,3)(5,8,11,7)(6,13,16,20)(9,23,25,35)(10,19,17,22)(12,27,29,15)(14,28,32,36)(18,24,21,26)(30,33,34,31);(1,3,4,2)(5,9,11,25)(6,14,16,32)(7,18,8,21)(10,26,17,24)(12,28,29,36)(13,30,20,34)(15,33,27,31)(19,23,22,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 375131);