#429996

order	512
nilpotency class	3
isoclinism	isoclinic class of #419461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1608 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)(6,12,15,24)(7,16)(13,26,28,17)(14,21,30,29)(18,31)(19,20)(27,36,33,32)(34,35);(5,6,11,12,10,15,9,24)(7,17,20,28,16,26,19,13)(8,21,25,30,23,29,22,14)(18,33,35,36,31,27,34,32);(5,7,10,16)(6,13,15,28)(8,18,23,31)(9,20,11,19)(12,26,24,17)(14,27,30,33)(21,36,29,32)(22,35,25,34);(2,4)(5,8,10,23)(6,14)(7,18,16,31)(9,22,11,25)(12,21)(13,27)(15,30)(17,32)(19,34,20,35)(24,29)(26,36)(28,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337192);