#358952

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

792 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)(7,9)(8,21)(10,15)(11,16)(12,18)(13,25)(14,28)(17,32)(19,26)(20,27)(22,29)(23,34)(24,30)(31,36)(33,35);(5,7)(6,12)(8,22,25,33)(9,15)(10,19)(11,20)(13,23,21,32)(14,27)(16,26)(17,24,34,36)(18,28)(29,31,35,30);(5,8,10,24,11,25,28,36)(6,13,14,30,16,21,15,31)(7,17,18,33,20,34,19,22)(9,23,26,35,27,32,12,29);(5,9,10,26,11,27,28,12)(6,7,15,19,16,20,14,18)(8,23,24,35,25,32,36,29)(13,17,31,22,21,34,30,33)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266148);