#7625693

order	512
nilpotency class	2
isoclinism	isoclinic class of #62842
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

10934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532889);