#7672208

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

7008 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579404);