#7712724

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

7338 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)(8,11)(15,18)(17,25)(20,22)(21,24);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14,19,16)(15,22,18,20)(17,21,25,24)(23,28,26,27);(6,10)(7,12)(13,15,19,18)(14,20,16,22)(17,26,25,23)(21,28,24,27);(13,16,19,14)(15,20,18,22)(17,24,25,21)(23,27,26,28);(5,7,9,12)(6,11,10,8)(13,17,19,25)(14,21,16,24)(15,23,18,26)(20,27,22,28);(5,8,9,11)(6,12,10,7)(13,18,19,15)(14,20,16,22)(17,26,25,23)(21,27,24,28)

Polycyclic presentation:

2^9|b^2=h,c^2=h,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, 7619920);