#417716

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60367	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2403 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)(7,18)(8,24)(9,16)(10,15)(11,17)(12,21)(13,25)(14,26)(19,29)(20,33)(22,31)(23,30)(27,32)(28,36)(34,35);(5,7,11,23)(6,12,17,31)(8,19,26,36)(9,22,10,21)(13,29,32,28)(14,33,24,34)(15,30,16,18)(20,27,35,25);(5,8)(6,13)(7,19)(9,25)(10,27)(11,26)(12,28)(14,16)(15,24)(17,32)(18,34)(20,22)(21,35)(23,36)(29,31)(30,33);(2,4)(5,8)(6,14)(7,20)(9,27)(10,25)(11,26)(12,29)(13,16)(15,32)(17,24)(18,34)(19,22)(21,36)(23,35)(28,31)(30,33)

Polycyclic presentation:

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

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, 324912);