#7625706

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×#3459	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7788 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^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, 7532902);