#366214

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1151 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)(5,6,11,16,9,17,10,15)(7,18,21,13,22,12,20,28)(8,23,25,30,26,14,24,31)(19,32,35,36,33,27,34,29);(1,3,4,2)(5,7,11,20,9,22,10,21)(6,12,15,28,17,18,16,13)(8,19,24,35,26,33,25,34)(14,27,31,36,23,32,30,29);(5,7)(6,13)(8,19)(9,22)(10,21)(11,20)(12,15)(14,29)(16,18)(17,28)(23,36)(24,35)(25,34)(26,33)(27,31)(30,32);(5,8,11,24,9,26,10,25)(6,14,15,31,17,23,16,30)(7,19,20,35,22,33,21,34)(12,27,28,36,18,32,13,29)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 273410);