#245991

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	#2631	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1010 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=i,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, 153187);