#7708173

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

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

6615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2,3,4)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20);(2,4)(21,22)(23,30)(24,34)(25,28)(26,29)(27,33)(31,35)(32,36);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23,26,33)(22,27,29,30)(24,32,25,31)(28,36,34,35);(7,14)(10,13)(15,20)(17,19)(21,24,26,25)(22,28,29,34)(23,31,33,32)(27,35,30,36);(6,11)(10,13)(12,18)(17,19)(21,25,26,24)(22,28,29,34)(23,32,33,31)(27,35,30,36);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7615369);