#108312

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

784 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15508);