#7748706

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

7626 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)(5,6)(7,11)(8,9)(10,12)(13,14)(15,18)(16,21)(17,20)(19,24)(22,25)(23,27)(26,28);(1,2,3,4)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26);(6,9)(10,11)(14,15)(17,25)(19,21)(23,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,21,24,19)(17,22,25,20)(23,27,28,26);(5,7)(6,10)(8,12)(9,11)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7655902);