#7678278

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

6940 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)(6,10)(11,12)(14,20)(18,22)(21,24)(26,28);(5,6)(7,11)(8,12)(9,10)(13,14,19,20)(15,24,16,21)(17,18,23,22)(25,26,27,28);(2,4)(13,15,19,16)(14,21,20,24)(17,27,23,25)(18,26,22,28);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16,19,15)(14,21,20,24)(17,27,23,25)(18,28,22,26);(13,17,19,23)(14,22,20,18)(15,25,16,27)(21,26,24,28);(5,8,9,7)(6,12,10,11)(13,18,19,22)(14,23,20,17)(15,26,16,28)(21,25,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585474);