#7786490

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

6996 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)(6,10)(7,11)(13,14,19,23)(15,20,24,28)(16,17,21,26)(18,22,27,25);(1,2,3,4)(6,10)(7,11)(13,15)(14,20)(16,18)(17,22)(19,24)(21,27)(23,28)(25,26);(5,6)(7,8)(9,10)(11,12)(15,24)(18,27)(20,28)(22,25);(6,10)(7,11)(13,16)(14,17)(15,18)(19,21)(20,22)(23,26)(24,27)(25,28);(5,7,9,11)(6,8,10,12)(13,17)(14,21)(15,25)(16,23)(18,20)(19,26)(22,24)(27,28);(5,8)(6,11)(7,10)(9,12)(13,18)(14,22)(15,16)(17,20)(19,27)(21,24)(23,25)(26,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7693686);