#7686138

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

6908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,e^2=gi,f^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, 7593334);