#407267

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efh,c^2=ef,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,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, 314463);