#7712782

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

7016 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619978);