#443898

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=egi,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=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, 351094);