#115850

order	512
nilpotency class	3
isoclinism	isoclinic class of #100003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 52 points:

(2,4)(5,6)(7,10)(8,15,16,11)(9,12)(13,17,20,19)(14,18)(21,22,26,32,27,30,25,31)(23,28,34,44,35,43,41,48)(24,36,39,46,40,29,38,47)(33,49,37,52,50,42,51,45);(1,2,3,4)(5,7)(6,10)(8,13)(9,14)(11,19)(12,18)(15,17)(16,20)(21,23)(22,28)(24,37)(25,41)(26,34)(27,35)(29,45)(30,43)(31,48)(32,44)(33,38)(36,52)(39,50)(40,51)(42,46)(47,49);(5,8,9,16)(6,11,12,15)(7,13,14,20)(10,17,18,19)(21,24,27,40)(22,29,30,36)(23,33,35,50)(25,39,26,38)(28,42,43,49)(31,47,32,46)(34,51,41,37)(44,52,48,45)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fhi,f^c=fhi,g^c=gh

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, 23046);