#7678269

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	C₂×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6708 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)(6,9)(10,12)(14,20)(18,28)(21,23)(25,27);(5,6,8,9)(7,10,11,12)(13,14)(15,23)(16,21)(17,28)(18,22)(19,20)(24,27)(25,26);(2,4)(13,15,19,16)(14,21,20,23)(17,26,22,24)(18,25,28,27);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16,19,15)(14,21,20,23)(17,26,22,24)(18,27,28,25);(13,17,19,22)(14,18,20,28)(15,24,16,26)(21,27,23,25);(13,18)(14,22)(15,25)(16,27)(17,20)(19,28)(21,24)(23,26)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585465);