#6424583

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2537	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,23,21,25,19,26)(20,28,30,32,29,24,27,31);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(20,29)(24,28)(27,30)(31,32);(9,11,12,16)(10,13,14,15)(17,19,21,22)(18,23,25,26)(20,30,29,27)(24,32,28,31);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,20,22,30,21,29,19,27)(18,24,23,31,25,28,26,32);(1,5,6,4)(2,7,3,8)(20,29)(24,28)(27,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi,d^c=di,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340641, #72340653, #72340675
maximal normal subgroup of	#72340641, #72340653, #72340675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6331779);