#7626417

order	512
nilpotency class	2
isoclinism	isoclinic class of #62862
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×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6772 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)(5,6)(7,11)(8,10)(9,12)(13,14,19,20)(15,16,17,21)(18,26,27,22)(23,28,25,24);(5,6,8,10)(7,9,12,11)(14,20)(15,17)(22,26)(23,25);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(13,15,19,17)(14,21,20,16)(18,23,27,25)(22,28,26,24);(5,7,8,12)(6,9,10,11)(13,16,19,21)(14,15,20,17)(18,24,27,28)(22,23,26,25);(13,17,19,15)(14,21,20,16)(18,25,27,23)(22,28,26,24);(13,18)(14,22)(15,23)(16,24)(17,25)(19,27)(20,26)(21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533613);