#7626269

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
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×#3513	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7282 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)(13,14,19,17)(15,16,22,20)(18,25,26,21)(23,27,28,24);(7,11)(10,12)(14,17)(15,22)(21,25)(23,28);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15,19,22)(14,20,17,16)(18,23,26,28)(21,27,25,24);(13,16)(14,15)(17,22)(18,24)(19,20)(21,23)(25,28)(26,27);(5,7,8,11)(6,10,9,12)(13,17,19,14)(15,16,22,20)(18,25,26,21)(23,24,28,27);(5,6)(7,10)(8,9)(11,12)(13,18,19,26)(14,21,17,25)(15,23,22,28)(16,24,20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533465);