#347051

order	512
nilpotency class	3
isoclinism	isoclinic class of #347042
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#62820	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,b^a=be,c^a=cf,d^a=dg,f^b=fh,g^b=gi,e^c=eh,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210515, #72234606, #72234613, …
maximal normal subgroup of	#72210515, #72234606, #72234613, …
maximal quotient of	#72341142

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254247);