#459752

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

921 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:

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

Polycyclic presentation:

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

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, 366948);