#365063

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1662 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)(7,9)(8,21)(10,29)(11,16)(12,17)(13,19)(14,25)(15,27)(18,30)(20,28)(22,23)(24,32)(26,31)(33,36)(34,35);(1,3,4,2)(5,7,10,19,12,20,11,15)(6,13,16,9,17,27,29,28)(8,22,24,18,26,33,25,34)(14,23,31,35,32,36,21,30);(5,8)(6,14)(7,18)(9,23)(10,24)(11,25)(12,26)(13,30)(15,22)(16,31)(17,32)(19,33)(20,34)(21,29)(27,35)(28,36);(5,9,10,27,12,28,11,13)(6,15,16,7,17,19,29,20)(8,23,24,35,26,36,25,30)(14,22,31,18,32,33,21,34)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,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, 272259);