#7642373

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

7382 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)(7,11)(10,12)(14,18)(15,21)(20,23)(22,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(13,14,17,18)(15,19,21,25)(16,23,24,20)(22,28,27,26);(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(5,6)(7,10)(8,9)(11,12)(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549569);