#7672206

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

7092 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)(6,10)(11,12)(13,14,19,17)(15,24,26,20)(16,22,27,21)(18,28,25,23);(5,6,9,10)(7,11,8,12)(13,15,19,26)(14,20,17,24)(16,25,27,18)(21,23,22,28);(2,3)(13,16,19,27)(14,21,17,22)(15,18,26,25)(20,28,24,23);(1,3,4,2)(5,7,9,8)(6,11,10,12)(13,16)(14,22)(15,25)(17,21)(18,26)(19,27)(20,28)(23,24);(13,17,19,14)(15,24,26,20)(16,21,27,22)(18,28,25,23);(5,8,9,7)(6,12,10,11)(13,18,19,25)(14,23,17,28)(15,16,26,27)(20,22,24,21)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=gh,e^2=i,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, 7579402);