#7812174

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

8364 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)(5,6,9,11)(7,8,12,10);(1,2)(3,4)(13,14)(15,20)(16,21)(17,23)(18,19)(22,27)(24,26)(25,28);(5,7,9,12)(6,10,11,8)(14,19)(17,27)(20,28)(21,24);(5,7)(6,8)(9,12)(10,11)(13,15)(14,20)(16,22)(17,24)(18,25)(19,28)(21,27)(23,26);(5,8)(6,7)(9,10)(11,12)(13,16)(14,21)(15,23)(17,20)(18,26)(19,24)(22,25)(27,28);(13,17)(14,22)(15,24)(16,20)(18,27)(19,23)(21,25)(26,28)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7719370);