#7850810

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

7299 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)(14,20)(15,17)(22,27)(23,25);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,16)(17,21)(18,22)(19,20)(23,24)(25,28)(26,27);(5,7)(6,9)(8,12)(10,11)(13,15,19,17)(14,21,20,16)(18,23,26,25)(22,28,27,24);(13,16,19,21)(14,15,20,17)(18,24,26,28)(22,23,27,25);(5,7)(6,10)(8,12)(9,11)(13,17,19,15)(14,21,20,16)(18,23,26,25)(22,24,27,28);(5,6)(7,9)(8,11)(10,12)(13,18)(14,22)(15,23)(16,24)(17,25)(19,26)(20,27)(21,28)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7758006);