#7686847

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

7014 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)(16,21)(17,22)(18,19)(23,25)(24,26)(27,28);(5,7,9,12)(6,10,8,11)(13,14)(15,20)(16,21)(17,22)(18,19)(23,25)(24,26)(27,28);(2,4)(5,8,9,6)(7,11,12,10)(14,19)(15,23)(21,24)(22,28);(1,2)(3,4)(13,15,18,23)(14,20,19,25)(16,22,26,28)(17,24,27,21);(13,16)(14,21)(15,22)(17,20)(18,26)(19,24)(23,28)(25,27);(13,17)(14,22)(15,24)(16,25)(18,27)(19,28)(20,26)(21,23)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7594043);