#7626319

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×#3540	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7244 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)(15,24)(16,17)(18,26)(19,20)(21,25)(22,28)(23,27);(7,11)(10,12)(14,20)(15,25)(17,22)(23,26);(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,26)(17,27)(18,22)(19,25)(20,24)(23,28);(1,2,4,3)(13,16)(14,22)(15,26)(17,20)(18,21)(19,28)(23,25)(24,27);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,17,19,22)(14,16,20,28)(15,27,25,18)(21,26,24,23);(5,6)(7,10)(8,9)(11,12)(13,18,19,27)(14,23,20,26)(15,22,25,17)(16,21,28,24)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=gi,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, 7533515);