#353932

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1930 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261128);