#7665722

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×#3482	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6847 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)(7,8)(9,12)(10,11)(15,21)(16,24)(18,22)(19,26);(1,2)(3,4)(5,7,10,9)(6,8,11,12)(13,14,17,20)(15,22,21,18)(16,19,24,26)(23,28,27,25);(6,11)(8,12)(13,15,17,21)(14,18,20,22)(16,23,24,27)(19,25,26,28);(13,16)(14,19)(15,23)(17,24)(18,25)(20,26)(21,27)(22,28);(5,8,10,12)(6,7,11,9);(5,9,10,7)(6,8,11,12)

Polycyclic presentation:

2^9|b^2=h,c^2=i,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, 7572918);