#7648474

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×#3475	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6813 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)(9,17)(12,19)(15,18)(16,20)(21,22,27,30)(23,29,26,28)(24,33,35,25)(31,32,34,36);(1,2)(3,4)(5,6)(7,14)(8,11)(9,12)(10,13)(15,20)(16,18)(17,19);(5,7,10,11)(6,8,13,14)(9,15,17,18)(12,16,19,20)(21,23,27,26)(22,28,30,29)(24,31,35,34)(25,32,33,36);(5,6)(7,8)(9,12)(10,13)(11,14)(15,16)(17,19)(18,20)(21,24,27,35)(22,25,30,33)(23,31,26,34)(28,32,29,36);(5,8)(6,11)(7,13)(9,16)(10,14)(12,18)(15,19)(17,20)(21,25)(22,24)(23,32)(26,36)(27,33)(28,31)(29,34)(30,35);(5,9,10,17)(6,12,13,19)(7,15,11,18)(8,16,14,20)(21,26,27,23)(22,29,30,28)(24,34,35,31)(25,36,33,32)

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7555670);