#7643519

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

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, 7550715);