#7658980

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

6924 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566176);