#7694178

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

7120 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)(5,6,9,8)(7,11,12,10)(13,14)(15,20)(16,25)(17,23)(18,24)(19,21)(22,26)(27,28);(5,7,9,12)(6,10,8,11)(13,15)(14,20)(16,17)(18,21)(19,24)(22,28)(23,25)(26,27);(2,4)(5,6,9,8)(7,10,12,11)(14,21)(15,24)(16,26)(23,28);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,16)(14,22)(15,17)(18,23)(19,26)(20,28)(21,25)(24,27);(5,8,9,6)(7,11,12,10)(13,17)(14,23)(15,16)(18,22)(19,27)(20,25)(21,28)(24,26);(13,18)(14,24)(15,21)(16,23)(17,25)(19,20)(22,27)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7601374);