#245986

order	512
nilpotency class	3
isoclinism	isoclinic class of #245903
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	C₂×C₄²	C₂⁴
prime radical	C₂×#562	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153182);