#505189

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

674 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)(7,12)(8,14)(9,13)(10,25)(11,16)(15,24)(17,28)(18,29)(19,30)(20,35)(21,31)(22,32)(23,33)(26,36)(27,34);(1,2,4,3)(5,7,17,21,9,22,11,27)(6,12,28,31,13,32,16,34)(8,19,24,18,23,20,10,26)(14,30,15,29,33,35,25,36);(1,2,4,3)(6,13)(7,18)(8,10)(11,17)(12,29)(14,15)(19,21)(20,27)(22,26)(23,24)(25,33)(30,31)(32,36)(34,35);(1,3,4,2)(5,8,11,10,9,23,17,24)(6,14,16,25,13,33,28,15)(7,19,27,26,22,20,21,18)(12,30,34,36,32,35,31,29)

Polycyclic presentation:

2^9|b^2=fh,c^2=e,d^2=fhi,g^2=h,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=ghi,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, 412385);