#433012

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

959 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)(5,6)(7,12,19,8)(9,14)(10,16)(11,15)(13,22,20,29)(17,27)(18,21,23,28)(24,32)(25,31)(26,33)(30,35,34,36);(1,3,4,2)(5,7,10,13,11,19,27,20)(6,12,17,29,15,8,16,22)(9,23,24,30,25,18,26,34)(14,28,33,36,31,21,32,35);(2,3)(5,8)(6,13,15,20)(7,17,19,16)(9,21)(10,22)(11,12)(14,30,31,34)(18,32,23,33)(24,35)(25,28)(26,36)(27,29);(5,9,11,25)(6,14,15,31)(7,18,19,23)(8,21,12,28)(10,26,27,24)(13,30,20,34)(16,33,17,32)(22,36,29,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340208);