#7716628

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

8102 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)(7,11)(10,12)(13,14,17,20)(15,18,23,21)(16,24,25,19)(22,27,28,26);(1,2)(3,4)(5,6,8,9)(7,10,11,12);(6,9)(10,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28);(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27);(5,7)(6,10)(8,11)(9,12)

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7623824);