#443902

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

776 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)(3,4)(5,6,10,15,11,16,21,32)(7,17,13,28,19,8,20,12)(9,23,25,35,26,14,29,31)(18,22,24,27,33,30,34,36);(5,7,10,20,11,19,21,13)(6,12,15,8,16,28,32,17)(9,24,25,18,26,34,29,33)(14,30,31,27,23,22,35,36);(5,8,21,28,11,17,10,12)(6,13,15,19,16,20,32,7)(9,22,29,36,26,30,25,27)(14,24,31,33,23,34,35,18);(2,4)(5,9,10,25,11,26,21,29)(6,14,15,31,16,23,32,35)(7,18,13,24,19,33,20,34)(8,22,12,27,17,30,28,36)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=egi,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, 351098);