#353912

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×#386
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1998 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)(7,16)(8,13)(9,22)(10,15)(11,14)(12,18)(17,32)(19,27)(20,30)(21,28)(23,26)(24,29)(25,33)(31,34)(35,36);(5,7,10,18,11,19,9,23)(6,12,15,27,14,26,22,16)(8,17,20,35,21,34,24,33)(13,25,30,32,28,36,29,31);(1,3,4,2)(5,8)(6,13)(7,17)(9,24)(10,20)(11,21)(12,25)(14,28)(15,30)(16,31)(18,35)(19,34)(22,29)(23,33)(26,36)(27,32);(6,14)(12,26)(13,28)(15,22)(16,27)(25,36)(29,30)(31,32)

Polycyclic presentation:

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

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, 261108);