#8071411

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

7580 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:

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

Polycyclic presentation:

2^9|a^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7978607);