#7737491

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

6990 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7644687);