#7876830

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

7119 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)(5,6,9,7)(8,11,12,10)(13,14)(15,20)(16,18)(17,26)(19,21)(22,24)(23,27)(25,28);(1,2)(3,4)(13,15,19,23)(14,20,21,27)(16,24,25,17)(18,22,28,26);(6,7)(10,11)(14,21)(15,23)(16,25)(22,26);(5,7,9,6)(8,11,12,10)(13,16)(14,18)(15,24)(17,23)(19,25)(20,22)(21,28)(26,27);(5,8,9,12)(6,10,7,11)(13,17,19,24)(14,22,21,26)(15,16,23,25)(18,20,28,27);(13,18)(14,16)(15,22)(17,27)(19,28)(20,24)(21,25)(23,26)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=hi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7784026);