#364178

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
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

1419 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=egi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^c=hi,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, 271374);