#504673

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13316	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411869);