#7655287

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×#3483	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7170 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)(13,14,19,17)(15,22,20,16)(18,21,26,25)(23,28,27,24);(1,2)(3,4)(5,6,10,9)(7,12,8,11);(5,7,10,8)(6,11,9,12)(13,15)(14,16)(17,22)(18,23)(19,20)(21,24)(25,28)(26,27);(13,16,19,22)(14,20,17,15)(18,24,26,28)(21,27,25,23);(5,8,10,7)(6,11,9,12)(13,17,19,14)(15,22,20,16)(18,21,26,25)(23,24,27,28);(5,9,10,6)(7,12,8,11)(13,18,19,26)(14,21,17,25)(15,23,20,27)(16,24,22,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,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, 7562483);