#412567

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

974 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=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, 319763);