#7682471

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

7203 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)(3,4)(6,9)(10,12)(13,14,17,18)(15,19,21,26)(16,20,25,23)(22,27,24,28);(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,23)(17,18)(20,25)(21,26)(22,28)(24,27);(2,4)(14,18)(15,21)(20,23)(22,24);(1,2)(3,4)(13,15)(14,19)(16,24)(17,21)(18,26)(20,28)(22,25)(23,27);(5,7,8,11)(6,10,9,12)(13,16,17,25)(14,20,18,23)(15,22,21,24)(19,27,26,28);(13,14)(15,19)(16,20)(17,18)(21,26)(22,27)(23,25)(24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589667);