#7691553

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

7231 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)(5,6,8,10)(7,9,11,12)(13,14,19,16)(15,20,23,24)(17,21,27,25)(18,22,28,26);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,24)(17,22)(18,25)(19,23)(21,28)(26,27);(2,4)(14,16)(15,23)(21,25)(22,26);(1,2)(3,4)(13,16,19,14)(15,24,23,20)(17,25,27,21)(18,26,28,22);(5,7,8,11)(6,9,10,12)(13,17)(14,21)(15,22)(16,25)(18,24)(19,27)(20,28)(23,26);(13,18)(14,22)(15,25)(16,26)(17,20)(19,28)(21,23)(24,27)

Polycyclic presentation:

2^9|a^2=gi,d^2=i,e^2=g,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, 7598749);