#367549

order	512
nilpotency class	3
isoclinism	isoclinic class of #25446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 274745);