#405772

order	512
nilpotency class	3
isoclinism	isoclinic class of #405772
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14918
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1973 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)(3,7,13,22)(4,14,17,8)(5,18,19,9)(6,10)(11,24)(12,26)(15,20)(16,28)(21,27)(23,30)(25,32,29,31);(1,3)(2,7,10,22)(4,15,17,11)(5,12)(6,13)(8,20)(9,21,18,26)(14,24)(16,29,30,25)(19,27)(23,31)(28,32);(1,4)(2,8)(3,11)(5,16)(6,17)(7,20)(9,23)(10,14)(12,25)(13,15)(18,28)(19,30)(21,31)(22,24)(26,32)(27,29);(1,5)(2,9)(3,12)(4,16)(6,19)(7,21)(8,23)(10,18)(11,25)(13,27)(14,28)(15,29)(17,30)(20,31)(22,26)(24,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=h,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,h^c=hi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237761, #72237762, #72237764, …
maximal normal subgroup of	#72237761, #72237762, #72237764, …
maximal quotient of	#72265877

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312968);