#411585

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	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1836 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)(7,17)(8,20)(9,10)(15,16)(18,32)(19,23)(21,22)(24,27)(25,29)(26,35)(34,36);(1,2)(3,4)(5,6)(7,13)(8,14)(9,15)(10,12)(11,16)(17,27)(18,26)(19,28)(20,29)(21,25)(22,30)(23,24)(31,35)(32,36)(33,34);(2,4)(5,7)(6,13)(8,18)(9,17)(10,23)(11,19)(12,24)(14,26)(15,27)(16,28)(20,31)(21,33)(22,32)(25,34)(29,35)(30,36);(5,8,9,20,11,22,10,21)(6,14,15,29,16,30,12,25)(7,18,17,31,19,32,23,33)(13,26,27,35,28,36,24,34)

Polycyclic presentation:

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

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, 318781);