#245653

order	512
nilpotency class	3
isoclinism	isoclinic class of #245615
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
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1123 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)(3,13)(4,8)(5,11)(6,10)(7,12)(9,14)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,33)(28,34)(29,35)(30,43)(31,38)(32,39)(36,41)(37,44)(40,47)(42,46)(45,48);(1,3,7,14)(2,8,12,16)(4,5,15,6)(9,11,13,10)(17,19,22,29)(18,23,26,35)(20,27,31,41)(21,28,32,42)(24,36,38,33)(25,34,39,46)(30,40,44,48)(37,47,43,45);(1,4)(2,9)(3,6)(5,14)(7,15)(8,10)(11,16)(12,13)(17,20,22,31)(18,24,26,38)(19,27,29,41)(21,30,32,44)(23,33,35,36)(25,37,39,43)(28,40,42,48)(34,45,46,47);(1,4)(2,9)(3,5)(6,14)(7,15)(8,11)(10,16)(12,13)(17,21)(18,25)(19,28)(20,30)(22,32)(23,34)(24,37)(26,39)(27,40)(29,42)(31,44)(33,45)(35,46)(36,47)(38,43)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341226
maximal normal subgroup of	#72341226

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152849);