#7779695

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

7299 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

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, 7686891);