#7626311

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×#3541	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7154 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)(13,14,19,24)(15,22,17,20)(16,18,23,21)(25,28,27,26);(7,11)(10,12)(13,15,19,17)(14,20,24,22)(16,27,23,25)(18,28,21,26);(5,6)(7,10)(8,9)(11,12)(13,16,19,23)(14,21,24,18)(15,25,17,27)(20,28,22,26);(1,2,4,3)(13,17,19,15)(14,22,24,20)(16,27,23,25)(18,28,21,26);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,18,19,21)(14,23,24,16)(15,26,17,28)(20,27,22,25);(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,22)(18,23)(19,24)(25,28)(26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=i,e^2=i,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, 7533507);