#7627472

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
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

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:

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

Polycyclic presentation:

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

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, 7534668);