#7689179

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

7165 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,20)(15,19,22,17)(16,21,25,27)(23,24,28,26);(1,2,3,4)(13,15)(14,19)(16,23)(17,20)(18,22)(21,24)(25,28)(26,27);(6,9)(10,12)(14,20)(15,22)(21,27)(23,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,23)(17,20)(18,22)(21,24)(25,28)(26,27);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,23)(17,26)(18,25)(19,24)(20,27)(22,28);(13,17,18,19)(14,15,20,22)(16,24,25,26)(21,28,27,23)

Polycyclic presentation:

2^9|a^2=i,b^2=g,e^2=hi,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, 7596375);