#186045

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2306 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)(3,4)(5,6)(7,13,15,10)(8,16,17,11)(9,12)(14,19)(18,20)(21,22,26,30,27,31,25,29)(23,32,33,24,28,35,34,36);(2,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,28)(22,24,31,36)(25,33,26,34)(29,35,30,32);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,24)(22,28)(23,31)(25,32)(26,35)(27,36)(29,33)(30,34);(6,12)(10,13)(11,16)(18,19)(21,25,27,26)(22,29,31,30)(23,33,28,34)(24,35,36,32)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93241);