#7681122

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

7065 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)(15,17)(16,21)(18,20)(19,22)(23,26)(24,27)(25,28);(5,6)(7,11)(8,12)(9,10)(13,15,18,22)(14,19,20,17)(16,24,25,23)(21,26,28,27);(2,3)(14,20)(15,22)(16,25)(26,27);(1,3,4,2)(5,7,9,8)(6,11,10,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,26)(25,28);(5,8,9,7)(6,12,10,11)(13,16,18,25)(14,21,20,28)(15,23,22,24)(17,26,19,27);(13,17,18,19)(14,22,20,15)(16,26,25,27)(21,24,28,23)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=gh,e^2=g,f^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, 7588318);