#411722

order	512
nilpotency class	3
isoclinism	isoclinic class of #411585
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1848 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:

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

Polycyclic presentation:

2^9|d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318918);