#7648473

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

7113 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)(5,6,8,9)(7,11,12,10)(17,25)(20,27)(23,26)(24,28);(1,2)(3,4)(6,9)(10,11)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,26)(25,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,19)(14,16,21,22)(17,23,25,26)(20,24,27,28);(5,7,8,12)(6,10,9,11)(13,14)(15,16)(17,20)(18,21)(19,22)(23,24)(25,27)(26,28);(13,16)(14,19)(15,21)(17,24)(18,22)(20,26)(23,27)(25,28);(13,17,18,25)(14,20,21,27)(15,23,19,26)(16,24,22,28)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7555669);