#462287

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
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

691 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)(5,6,11,16)(7,18)(8,13)(9,14,26,31)(10,15,22,17)(12,21)(19,33)(20,27)(23,28)(24,29)(25,30,35,32)(34,36);(5,7,10,20,11,13,22,21)(6,12,17,8,16,27,15,18)(9,19,25,34,26,28,35,29)(14,24,32,23,31,36,30,33);(5,8,22,12,11,18,10,27)(6,13,15,20,16,7,17,21)(9,24,35,33,26,36,25,23)(14,29,30,28,31,34,32,19);(2,4)(5,9,10,25,11,26,22,35)(6,14,15,30,16,31,17,32)(7,19,20,34,13,28,21,29)(8,23,12,24,18,33,27,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369483);