#7645760

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

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

8182 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)(7,11)(13,14,18,22)(15,20,16,19)(17,21,25,28)(23,27,24,26);(1,2)(3,4)(5,6,8,9)(7,10,11,12);(5,6)(7,10)(8,9)(11,12)(13,15,18,16)(14,19,22,20)(17,23,25,24)(21,26,28,27);(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,20,22,19)(17,23,25,24)(21,26,28,27);(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28);(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=h,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, 7552956);