#246338

order	512
nilpotency class	3
isoclinism	isoclinic class of #246319
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	#60568	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1148 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153534);