#186056

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
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×#3535	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2394 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)(21,22,27,30,26,23,25,29)(24,28,32,35,34,31,33,36);(2,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,26,22)(24,31,34,28)(25,30,27,29)(32,36,33,35);(5,8)(6,11)(7,14)(9,17)(10,18)(12,16)(13,19)(15,20)(21,24)(22,28)(23,31)(25,32)(26,34)(27,33)(29,35)(30,36);(8,17)(11,16)(14,20)(18,19)(21,25,26,27)(22,29,23,30)(24,32,34,33)(28,35,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93252);