#7678281

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

6800 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)(5,6,9,8)(7,11,12,10)(13,14,19,16)(15,23,26,20)(17,21,27,25)(18,24,28,22);(5,7,9,12)(6,10,8,11)(13,15)(14,20)(16,23)(17,22)(18,25)(19,26)(21,28)(24,27);(2,4)(13,14,19,16)(15,23,26,20)(17,25,27,21)(18,22,28,24);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,16,19,14)(15,23,26,20)(17,25,27,21)(18,24,28,22);(13,17,19,27)(14,21,16,25)(15,24,26,22)(18,23,28,20);(5,8,9,6)(7,11,12,10)(13,18,19,28)(14,22,16,24)(15,25,26,21)(17,23,27,20)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=g,e^2=i,f^2=g,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, 7585477);