#245776

order	512
nilpotency class	3
isoclinism	isoclinic class of #245711
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	#3506	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1041 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)(3,12)(4,8)(5,9)(6,11)(7,13)(10,15)(14,16)(17,18)(19,30)(20,35)(21,38)(22,28)(23,27)(24,29)(25,33)(26,37)(31,40)(32,41)(34,44)(36,43)(39,42)(45,47)(46,48);(1,3,6,4)(2,7,11,16)(5,14,15,13)(8,9,12,10)(17,19)(18,25)(20,31)(21,32)(22,39)(23,33)(24,34)(26,41)(27,44)(28,30)(29,42)(35,45)(36,48)(37,47)(38,46)(40,43);(1,4,6,3)(2,8,11,12)(5,14,15,13)(7,9,16,10)(17,20,24,21)(18,26,29,43)(19,31,34,32)(22,37,23,36)(25,40,42,41)(27,35,28,38)(30,45,44,46)(33,48,39,47);(2,9)(5,15)(7,12)(8,16)(10,11)(13,14)(17,21,24,20)(18,26,29,43)(19,32,34,31)(22,36,23,37)(25,41,42,40)(27,35,28,38)(30,46,44,45)(33,47,39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152972);