#7712006

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

7206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

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, 7619202);