#7625723

order	512
nilpotency class	2
isoclinism	isoclinic class of #62842
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×#537	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7108 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)(10,12)(13,14,20,24)(15,25,17,21)(16,27,18,22)(19,26,28,23);(5,6)(7,10)(8,9)(11,12)(13,15,20,17)(14,21,24,25)(16,26,18,23)(19,22,28,27);(1,3,4,2)(13,16,20,18)(14,22,24,27)(15,26,17,23)(19,21,28,25);(5,7,8,11)(6,10,9,12)(13,17,20,15)(14,21,24,25)(16,26,18,23)(19,27,28,22);(13,18,20,16)(14,22,24,27)(15,26,17,23)(19,25,28,21);(13,19)(14,23)(15,22)(16,21)(17,27)(18,25)(20,28)(24,26)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^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, 7532919);