#7653848

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×#3522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7336 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)(5,6,10,9)(7,12,8,11);(1,2,3,4)(14,18)(19,25)(20,26)(27,28);(5,7,10,8)(6,11,9,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,28)(24,26);(5,7,10,8)(6,11,9,12);(5,8,10,7)(6,11,9,12)(13,15,17,22)(14,19,18,25)(16,23,24,21)(20,28,26,27);(5,9,10,6)(7,12,8,11)(13,16,17,24)(14,20,18,26)(15,21,22,23)(19,27,25,28)

Polycyclic presentation:

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

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, 7561044);