#7679240

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

7039 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7586436);