#346426

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2290 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)(6,11)(8,9)(12,18)(13,17)(14,16)(19,20)(21,22,25,28)(23,26,30,33)(24,27,31,34)(29,35,36,32);(5,6)(7,12)(8,11)(9,17)(10,13)(14,19)(15,20)(16,18)(21,23)(22,26)(24,29)(25,30)(27,32)(28,33)(31,36)(34,35);(5,7,10,15)(6,12,13,20)(8,16,9,14)(11,18,17,19)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,36)(33,35);(2,4)(5,8,10,9)(6,11,13,17)(7,14,15,16)(12,19,20,18)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253622);