#7732487

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

7516 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)(5,6,10,9)(7,12,8,11)(13,14,18,19)(15,20,22,17)(16,24,26,21)(23,25,28,27);(1,2,3,4)(5,7,10,8)(6,11,9,12);(7,8)(11,12)(14,19)(15,22)(21,24)(23,28);(13,15)(14,20)(16,23)(17,19)(18,22)(21,27)(24,25)(26,28);(5,8,10,7)(6,11,9,12)(13,16,18,26)(14,21,19,24)(15,23,22,28)(17,25,20,27);(5,9,10,6)(7,11,8,12)(13,17,18,20)(14,15,19,22)(16,25,26,27)(21,23,24,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7639683);