#7656680

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×#3516	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6452 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7563876);