#7648455

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

7334 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)(7,11)(8,10)(9,12)(13,14,19,23)(15,22,18,20)(16,17,27,21)(24,28,26,25);(1,2)(3,4)(5,7,10,9)(6,11,8,12)(13,15,19,18)(14,20,23,22)(16,24,27,26)(17,25,21,28);(5,8,10,6)(7,11,9,12)(13,16)(14,21)(15,24)(17,23)(18,26)(19,27)(20,28)(22,25);(5,9,10,7)(6,11,8,12)(13,17,19,21)(14,16,23,27)(15,25,18,28)(20,24,22,26);(13,16)(14,17)(15,24)(18,26)(19,27)(20,25)(21,23)(22,28);(13,18,19,15)(14,22,23,20)(16,26,27,24)(17,28,21,25)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7555651);