#7649127

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

7698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7556323);