#7738916

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

7422 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)(15,24)(16,20)(17,21)(18,22)(19,23)(25,27)(26,28);(1,2,3,4)(13,15,19,16)(14,20,23,24)(17,22,27,26)(18,21,28,25);(6,9)(10,11)(15,16)(17,27)(20,24)(21,25);(5,6)(7,10)(8,9)(11,12)(13,16,19,15)(14,20,23,24)(17,26,27,22)(18,21,28,25);(5,7)(6,10)(8,12)(9,11)(13,17)(14,21)(15,22)(16,26)(18,24)(19,27)(20,28)(23,25);(13,18,19,28)(14,22,23,26)(15,25,16,21)(17,24,27,20)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7646112);