#246407

order	512
nilpotency class	3
isoclinism	isoclinic class of #246367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,7)(5,9)(6,11)(8,14)(12,16)(13,15)(17,18,22,26)(19,23,29,36)(20,24,31,38)(21,32,33,25)(27,39,41,34)(28,42,43,35)(30,37,44,45)(40,47,48,46);(1,3)(2,6)(4,13)(5,12)(7,10)(8,16)(9,15)(11,14)(17,19,22,29)(18,23,26,36)(20,27,31,41)(21,28,33,43)(24,34,38,39)(25,35,32,42)(30,40,44,48)(37,46,45,47);(6,15)(10,16)(17,20,22,31)(18,24,26,38)(19,27,29,41)(21,30,33,44)(23,34,36,39)(25,37,32,45)(28,40,43,48)(35,46,42,47);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(17,21)(18,25)(19,28)(20,30)(22,33)(23,35)(24,37)(26,32)(27,40)(29,43)(31,44)(34,46)(36,42)(38,45)(39,47)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153603);