#7850266

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

7129 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)(13,14,20,24)(15,21,17,19)(16,26,28,22)(18,25,27,23);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,20,17)(14,21,24,19)(16,23,28,25)(18,22,27,26);(5,7)(6,9)(8,12)(10,11)(13,16,20,28)(14,22,24,26)(15,25,17,23)(18,19,27,21);(13,17,20,15)(14,19,24,21)(16,25,28,23)(18,26,27,22);(5,7)(6,10)(8,12)(9,11)(13,18)(14,23)(15,22)(16,19)(17,26)(20,27)(21,28)(24,25);(5,6)(7,9)(8,11)(10,12)(13,19)(14,15)(16,27)(17,24)(18,28)(20,21)(22,25)(23,26)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^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, 7757462);