#7682477

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

6924 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)(6,10)(11,12)(13,14,19,17)(15,23,24,20)(16,21,26,27)(18,28,25,22);(5,6)(7,11)(8,12)(9,10)(13,15,19,24)(14,20,17,23)(16,25,26,18)(21,28,27,22);(2,4)(15,24)(18,25)(20,23)(22,28);(1,2)(3,4)(13,16)(14,21)(15,25)(17,27)(18,24)(19,26)(20,28)(22,23);(5,7,9,8)(6,11,10,12)(13,17,19,14)(15,20,24,23)(16,21,26,27)(18,28,25,22);(5,8,9,7)(6,12,10,11)(13,18,19,25)(14,22,17,28)(15,16,24,26)(20,21,23,27)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=g,f^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, 7589673);