#7626277

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3522	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7158 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533473);