#7879287

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

7171 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)(13,14)(15,20)(16,17)(18,21)(19,23)(22,28)(24,25)(26,27);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,19,18)(14,20,23,21)(16,24,22,26)(17,25,28,27);(5,7)(6,8)(9,11)(10,12)(13,15,19,18)(14,21,23,20)(16,26,22,24)(17,25,28,27);(13,16,19,22)(14,17,23,28)(15,24,18,26)(20,25,21,27);(5,7)(6,10)(8,12)(9,11)(13,17)(14,22)(15,25)(16,23)(18,27)(19,28)(20,26)(21,24);(5,8,9,10)(6,11,12,7)(13,18,19,15)(14,21,23,20)(16,26,22,24)(17,27,28,25)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7786483);