#505191

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×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=ei,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, 412387);