#114535

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	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=i,g^2=h,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, 21731);