#7646936

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

8544 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)(7,11,12,9)(13,14)(15,20)(16,24)(17,18)(19,22)(21,27)(23,26)(25,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,19,17)(14,20,22,18)(16,23,27,25)(21,28,24,26);(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,23)(17,25)(18,26)(19,27)(20,28)(22,24);(5,6)(7,9)(8,10)(11,12)(13,17,19,15)(14,20,22,18)(16,25,27,23)(21,28,24,26);(13,18,19,20)(14,17,22,15)(16,26,27,28)(21,25,24,23);(13,16)(14,21)(15,23)(17,25)(18,26)(19,27)(20,28)(22,24)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^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, 7554132);