#443901

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=eg,d^2=e,e^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,g^b=gi,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, 351097);