#7665650

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×#3480	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

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, 7572846);