#7632264

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

7622 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)(6,9)(10,12)(13,14)(15,16)(17,20)(18,21)(19,22)(23,24)(25,27)(26,28);(1,2,3,4)(13,15,19,17)(14,20,22,16)(18,25,26,23)(21,24,28,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,17)(15,22)(18,24)(19,20)(21,25)(23,28)(26,27);(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,22)(18,24)(19,20)(21,23)(25,28)(26,27);(13,17,19,15)(14,20,22,16)(18,25,26,23)(21,27,28,24);(13,18)(14,21)(15,23)(16,24)(17,25)(19,26)(20,27)(22,28)

Polycyclic presentation:

2^9|b^2=g,d^2=hi,e^2=i,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, 7539460);