#7633305

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

8178 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)(6,9)(10,12)(13,14)(15,24)(16,17)(18,22)(19,23)(20,25)(21,27)(26,28);(1,2)(3,4)(13,15,19,25)(14,20,23,24)(16,26,27,22)(17,18,21,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,27)(14,21,23,17)(15,22,25,26)(18,24,28,20);(13,17)(14,16)(15,18)(19,21)(20,26)(22,24)(23,27)(25,28);(13,14)(15,20)(16,21)(17,27)(18,22)(19,23)(24,25)(26,28);(5,7,8,11)(6,10,9,12)(13,18)(14,22)(15,17)(16,24)(19,28)(20,27)(21,25)(23,26)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,f^2=hi,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, 7540501);