#7631855

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

8267 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539051);