#7650830

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×#3506	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6766 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,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, 7558026);