#7718162

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

8160 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)(8,12)(10,11)(14,15)(17,25)(19,21)(23,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,15)(16,19,24,21)(17,20,25,22)(23,26,28,27);(6,7)(10,11)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27);(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27);(5,7,9,6)(8,11,12,10)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28);(5,8)(6,10)(7,11)(9,12)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

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, 7625358);