#428399

order	512
nilpotency class	3
isoclinism	isoclinic class of #428399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15383
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1219 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)(3,11,14,7)(4,15,17,8)(5,9,19,23)(6,10)(12,24)(13,25)(16,28)(18,21)(20,27)(22,29)(26,32,30,31);(1,3)(2,7)(4,12)(5,18)(6,14)(8,20)(9,21)(10,11)(13,19)(15,24)(16,26)(17,27)(22,32)(23,25)(28,31)(29,30);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,15)(11,24)(13,26)(14,27)(18,30)(19,29)(21,31)(23,28)(25,32);(1,5)(2,9,10,23)(3,13,14,18)(4,16)(6,19)(7,21)(8,22,15,28)(11,25)(12,26,27,30)(17,29)(20,31)(24,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237787, #72237790, #72237795, …
maximal normal subgroup of	#72237787, #72237790, #72237795, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335595);