#6458393

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293017, #72293021, #72293074
maximal normal subgroup of	#72293017, #72293021, #72293074

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365589);