#7674375

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

6830 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,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, 7581571);