#417701

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
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2333 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,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, 324897);