#7714883

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

7106 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)(7,8)(9,10)(11,12)(13,14)(15,21)(16,24)(17,18)(19,22)(20,25)(23,26)(27,28);(1,2,3,4)(13,14,17,18)(15,19,22,21)(16,20,25,24)(23,26,28,27);(6,10)(7,11)(14,18)(15,22)(20,24)(23,28);(5,7,9,11)(6,8,10,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,26)(24,25)(27,28);(5,8)(6,11)(7,10)(9,12)(13,15,17,22)(14,19,18,21)(16,23,25,28)(20,26,24,27);(13,16)(14,20)(15,23)(17,25)(18,24)(19,26)(21,27)(22,28)

Polycyclic presentation:

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

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, 7622079);