#186048

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×#3536	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2602 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)(3,4)(5,6)(7,13,15,10)(8,16,17,11)(9,12)(14,19)(18,20)(21,22,25,27,26,30,24,29)(23,28,33,34,31,35,32,36);(2,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(22,27)(24,25)(28,34)(29,30)(32,33)(35,36);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,23)(22,28)(24,32)(25,33)(26,31)(27,34)(29,36)(30,35);(8,17)(11,16)(14,20)(18,19)(23,31)(28,35)(32,33)(34,36)

Polycyclic presentation:

2^9|a^2=g,c^2=h,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, 93244);