#7627611

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	C₂²×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6230 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)(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(21,22)(23,32)(24,35)(25,28)(26,29)(27,31)(30,38)(33,42)(34,45)(36,46)(37,48)(39,41)(40,43)(44,50)(47,51)(49,52);(6,7)(11,13)(12,14)(18,19)(21,23,28,26)(22,29,25,32)(24,33,38,36)(27,34,41,40)(30,42,35,46)(31,43,39,45)(37,47,50,49)(44,51,48,52);(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,24)(22,30)(23,33)(25,35)(26,36)(27,37)(28,38)(29,42)(31,44)(32,46)(34,47)(39,48)(40,49)(41,50)(43,51)(45,52);(5,8)(6,11)(7,13)(9,15)(10,16)(12,18)(14,19)(17,20)(21,25,28,22)(23,29,26,32)(24,35,38,30)(27,39,41,31)(33,42,36,46)(34,43,40,45)(37,48,50,44)(47,51,49,52);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,26,28,23)(22,29,25,32)(24,36,38,33)(27,34,41,40)(30,42,35,46)(31,45,39,43)(37,47,50,49)(44,52,48,51);(21,27)(22,31)(23,34)(24,37)(25,39)(26,40)(28,41)(29,43)(30,44)(32,45)(33,47)(35,48)(36,49)(38,50)(42,51)(46,52)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=h,e^2=g,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, 7534807);