#7626447

order	512
nilpotency class	2
isoclinism	isoclinic class of #62862
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533643);