#7659046

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

7360 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)(6,8)(7,11)(14,20)(16,26)(21,23)(24,27);(1,2)(3,4)(5,6,9,8)(7,12,11,10)(13,14)(15,23)(16,18)(17,21)(19,20)(22,26)(24,28)(25,27);(5,7,9,11)(6,10,8,12)(13,15,19,17)(14,21,20,23)(16,24,26,27)(18,25,22,28);(5,8,9,6)(7,12,11,10)(13,16)(14,22)(15,24)(17,27)(18,20)(19,26)(21,28)(23,25);(13,17,19,15)(14,21,20,23)(16,27,26,24)(18,25,22,28);(13,18,19,22)(14,16,20,26)(15,25,17,28)(21,24,23,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,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, 7566242);