#7650831

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×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6413 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)(5,6,11,13)(7,12,8,17)(9,10,14,20)(15,16,18,19)(21,22,26,25)(23,27,29,31)(24,28,32,33)(30,34,35,36);(1,2,3,4)(5,7,11,8)(6,12,13,17)(9,15,14,18)(10,16,20,19)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36);(6,13)(10,20)(12,17)(16,19)(23,29)(27,31)(30,35)(34,36);(5,8,11,7)(6,12,13,17)(9,15,14,18)(10,19,20,16)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,9,11,14)(6,10,13,20)(7,15,8,18)(12,16,17,19)(21,25,26,22)(23,31,29,27)(24,28,32,33)(30,34,35,36);(5,10)(6,14)(7,16)(8,19)(9,13)(11,20)(12,18)(15,17)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7558027);