#7732843

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

7200 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7640039);