#7632629

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

7647 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=i,e^2=i,f^2=h,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, 7539825);