#7678618

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

7106 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)(3,4)(5,6,9,8)(7,11,12,10)(17,25)(20,24)(22,28)(26,27);(5,7,9,12)(6,10,8,11)(13,14)(15,19)(16,18)(17,20)(21,23)(22,26)(24,25)(27,28);(2,4)(5,8,9,6)(7,11,12,10)(14,16)(19,21)(20,24)(26,27);(1,2)(3,4)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27);(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,27,28,26);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=i,e^2=i,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, 7585814);