#460563

order	512
nilpotency class	3
isoclinism	isoclinic class of #456658
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

972 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)(5,6,9,14,11,15,10,16)(7,12,19,29,18,28,25,27)(8,20,22,32,24,13,23,31)(17,33,21,36,34,26,35,30);(5,7)(6,12,15,28)(8,21)(9,25)(10,19)(11,18)(13,30,20,36)(14,27,16,29)(17,22)(23,34)(24,35)(26,32,33,31);(5,8,11,24)(6,13,15,20)(7,17,18,34)(9,23,10,22)(12,26,28,33)(14,32,16,31)(19,35,25,21)(27,30,29,36);(2,4)(6,14)(9,10)(12,27)(13,31)(15,16)(17,34)(19,25)(20,32)(22,23)(26,30)(28,29)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285470, #72285507, #72285571
maximal normal subgroup of	#72285470, #72285507, #72285571

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367759);