#7810591

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

8742 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)(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,24)(17,19)(20,22)(21,26)(23,28)(25,27);(1,2)(3,4)(13,15,17,22)(14,18,19,20)(16,25,26,23)(21,28,24,27);(6,10)(7,11)(14,19)(15,22)(21,24)(23,25);(5,7,9,11)(6,8,10,12)(13,15)(14,20)(16,23)(17,22)(18,19)(21,28)(24,27)(25,26);(5,8)(6,11)(7,10)(9,12)(13,16,17,26)(14,21,19,24)(15,23,22,25)(18,27,20,28);(13,14)(15,20)(16,21)(17,19)(18,22)(23,28)(24,26)(25,27)

Polycyclic presentation:

2^9|b^2=i,d^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72298998, #72299024, #72299033, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7717787);