#272402

order	512
nilpotency class	3
isoclinism	isoclinic class of #272283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

816 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)(3,13)(4,11)(5,8)(6,9)(7,12)(10,14)(15,16)(17,18,21,26,23,28,22,27)(19,24,30,39,32,41,31,40)(20,33,37,25,36,44,35,43)(29,45,34,38,47,42,46,48);(1,3,7,14)(2,8,12,16)(4,5,6,15)(9,13,11,10)(17,19)(18,24)(20,34)(21,30)(22,31)(23,32)(25,42)(26,39)(27,40)(28,41)(29,35)(33,38)(36,46)(37,47)(43,45)(44,48);(1,4,7,6)(2,9,12,11)(3,5,14,15)(8,13,16,10)(17,20)(18,25)(19,29)(21,35)(22,37)(23,36)(24,38)(26,33)(27,44)(28,43)(30,46)(31,34)(32,47)(39,45)(40,42)(41,48);(1,5,7,15)(2,10,12,13)(3,4,14,6)(8,11,16,9)(17,21,23,22)(18,26,28,27)(19,30,32,31)(20,35,36,37)(24,39,41,40)(25,33,43,44)(29,46,47,34)(38,45,48,42)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179598);