#353664

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
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×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2038 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:

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

Polycyclic presentation:

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

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, 260860);