#357378

order	512
nilpotency class	3
isoclinism	isoclinic class of #23360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1079 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)(5,6,11,15,10,16,21,27)(7,13,18,29,19,12,8,20)(9,23,26,14,24,31,25,32)(17,33,34,30,35,36,22,28);(5,7)(6,12,16,13)(8,11)(9,17,24,35)(10,19)(14,30)(15,29,27,20)(18,21)(22,25,34,26)(23,36)(28,32)(31,33);(1,3,4,2)(5,8,21,7,10,18,11,19)(6,13,15,20,16,12,27,29)(9,22,25,17,24,34,26,35)(14,30,31,33,32,28,23,36);(5,9,10,24)(6,14,16,32)(7,17,19,35)(8,22,18,34)(11,26,21,25)(12,28,13,30)(15,31,27,23)(20,33,29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,h^a=hi,c^b=cef,d^b=de,e^b=ei,g^b=gi,e^c=ei,g^c=gi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264574);