#7685127

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

8543 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)(6,9)(10,11)(13,14)(15,23)(16,18)(17,22)(19,21)(20,24)(25,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,17)(18,25)(19,24)(21,23)(22,28)(26,27);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(14,21)(15,24)(16,26)(22,25);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,16)(14,18)(15,17)(19,26)(20,25)(21,28)(22,23)(24,27);(13,17)(14,22)(15,16)(18,23)(19,27)(20,28)(21,25)(24,26);(13,18)(14,16)(15,25)(17,20)(19,28)(21,26)(22,24)(23,27)

Polycyclic presentation:

2^9|c^2=gh,d^2=gi,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, 7592323);