#7793447

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

7465 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)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(6,9)(10,11)(13,14,17,20)(15,18,23,21)(16,19,24,26)(22,25,28,27);(7,12)(10,11)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28);(5,7,8,12)(6,10,9,11)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7700643);