#7626515

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	C₂×#121	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6120 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)(6,8)(11,13)(12,16)(18,19)(22,27)(25,37)(28,31)(29,38)(33,47)(36,48)(40,43)(46,52);(5,6,10,8)(7,13,15,11)(9,12,17,16)(14,19,20,18)(24,35)(29,38)(32,45)(36,48)(40,43)(42,49)(46,52)(50,51);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,22,26,27)(23,31,34,28)(24,29,35,38)(25,30,37,39)(32,43,45,40)(33,44,47,41)(36,42,48,49)(46,51,52,50);(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,19,20,18)(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);(1,3,4,2)(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);(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19)(21,25)(22,30)(23,33)(24,36)(26,37)(27,39)(28,41)(29,42)(31,44)(32,46)(34,47)(35,48)(38,49)(40,50)(43,51)(45,52)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=gh,e^2=i,f^2=g,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, 7533711);