#7987102

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

6626 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)(7,16)(8,10)(9,20)(11,13)(12,17)(14,19)(15,18)(21,22,26,25)(23,29,31,27)(24,28,33,32)(30,35,36,34);(1,2)(3,4)(5,7,11,17)(6,12,13,16)(8,15,19,20)(9,14,18,10)(22,25)(27,29)(28,32)(34,35);(6,13)(7,17)(8,19)(9,18)(21,22,26,25)(23,27,31,29)(24,32,33,28)(30,35,36,34);(5,8)(6,14)(7,15)(9,16)(10,13)(11,19)(12,18)(17,20)(21,23,26,31)(22,27,25,29)(24,30,33,36)(28,34,32,35);(5,9)(6,15)(7,14)(8,16)(10,17)(11,18)(12,19)(13,20)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36);(5,10,11,14)(6,8,13,19)(7,18,17,9)(12,20,16,15)(21,25,26,22)(23,29,31,27)(24,28,33,32)(30,34,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7894298);