#7625690

order	512
nilpotency class	2
isoclinism	isoclinic class of #62842
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

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

8098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,e^2=g,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, 7532886);