#7819915

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

7071 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)(15,24)(17,27)(20,25)(21,26);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,25)(16,18)(17,21)(19,23)(20,24)(22,28)(26,27);(6,9)(10,12)(13,15,19,24)(14,20,23,25)(16,27,22,17)(18,21,28,26);(7,11)(10,12)(13,16,19,22)(14,18,23,28)(15,17,24,27)(20,26,25,21);(5,7)(6,10)(8,11)(9,12)(13,17)(14,21)(15,16)(18,25)(19,27)(20,28)(22,24)(23,26);(5,6)(7,10)(8,9)(11,12)(13,18)(14,22)(15,26)(16,23)(17,25)(19,28)(20,27)(21,24)

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7727111);