#7686445

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
prime radical	#36103	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6937 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,25)(24,27)(26,28);(1,2)(3,4)(13,15,18,20)(14,19,22,16)(17,23,26,27)(21,25,28,24);(6,9)(10,12)(13,14,18,22)(15,16,20,19)(17,21,26,28)(23,24,27,25);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,25)(24,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,20,22,15)(17,25,26,24)(21,23,28,27);(13,17)(14,21)(15,23)(16,24)(18,26)(19,25)(20,27)(22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7593641);