#505188

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

661 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)(8,13)(9,23)(10,15)(11,16)(14,21)(17,33)(18,28)(19,29)(20,31)(22,32)(24,27)(25,34)(26,35)(30,36);(1,2,4,3)(5,7,10,25,11,19,26,20)(6,12,15,34,16,29,35,31)(8,18,21,17,22,36,9,24)(13,28,14,33,32,30,23,27);(1,2,4,3)(7,17)(8,21)(9,22)(10,26)(12,27)(13,14)(15,35)(18,25)(19,24)(20,36)(23,32)(28,31)(29,33)(30,34);(1,3,4,2)(5,8,10,21,11,22,26,9)(6,13,15,14,16,32,35,23)(7,18,25,17,19,36,20,24)(12,28,34,33,29,30,31,27)

Polycyclic presentation:

2^9|b^2=fh,c^2=e,d^2=fh,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, 412384);