#7643311

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
prime radical	C₂×#3494	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7246 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)(7,11)(10,12)(13,14)(15,21)(16,24)(17,19)(18,22)(20,26)(23,27)(25,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,19,22,21)(16,23,26,25)(20,27,24,28);(6,9)(10,12)(13,14,18,22)(15,19,17,21)(16,20,26,24)(23,27,25,28);(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,28)(22,24);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,26)(25,28);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,21,22,19)(16,25,26,23)(20,28,24,27)

Polycyclic presentation:

2^9|b^2=h,c^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550507);