#7741178

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

8246 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)(5,6)(7,11)(8,9)(10,12)(15,21)(19,25)(22,27)(26,28);(1,2)(3,4)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27);(6,9)(10,11)(14,18)(15,21)(20,23)(22,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,19,18,25)(16,22,24,27)(20,26,23,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(13,16)(14,20)(15,22)(17,24)(18,23)(19,26)(21,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7648374);