#7710065

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6856 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)(8,15)(12,17)(13,19)(18,20)(21,22,27,28)(23,25,32,29)(24,30,35,33)(26,34,36,31);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(22,28)(23,32)(30,33)(31,34);(6,10)(11,16)(12,17)(18,20)(21,23,27,32)(22,29,28,25)(24,34,35,31)(26,33,36,30);(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,24,27,35)(22,30,28,33)(23,31,32,34)(25,26,29,36);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,25,27,29)(22,23,28,32)(24,26,35,36)(30,31,33,34);(5,8)(6,12)(7,13)(9,15)(10,17)(11,18)(14,19)(16,20)(21,26)(22,31)(23,33)(24,29)(25,35)(27,36)(28,34)(30,32)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

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, 7617261);