#114538

order	512
nilpotency class	3
isoclinism	isoclinic class of #100003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

955 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dhi,f^c=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, 21734);