#353934

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	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1890 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:

(1,2,4,3)(5,6,9,16,11,14,10,15)(7,17,20,27,21,28,19,12)(8,22,23,29,25,30,24,13)(18,32,34,36,35,26,33,31);(5,7,10,19,11,21,9,20)(6,12,15,28,14,27,16,17)(8,18,24,33,25,35,23,34)(13,26,30,36,29,32,22,31);(1,3,4,2)(5,8,9,23,11,25,10,24)(6,13,16,22,14,29,15,30)(7,18,20,34,21,35,19,33)(12,26,17,31,27,32,28,36);(6,14)(12,27)(13,29)(15,16)(17,28)(22,30)(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,h^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, 261130);