#401480

order	512
nilpotency class	3
isoclinism	isoclinic class of #401480
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14817
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

1214 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)(4,8,16,23)(5,17)(6,10)(9,18)(11,19,14,27)(12,25)(13,21)(15,29,30,22)(20,26)(24,32,28,31);(1,3)(2,7)(4,14,16,11)(5,12)(6,13)(8,19,23,27)(9,20)(10,21)(15,28,30,24)(17,25)(18,26)(22,31,29,32);(1,4)(2,8,10,23)(3,11)(5,15)(6,16)(7,19,21,27)(9,22,17,29)(12,24)(13,14)(18,30)(20,31,25,32)(26,28);(1,5)(2,9,10,17)(3,12)(4,15)(6,18)(7,20,21,25)(8,22,23,29)(11,24)(13,26)(14,28)(16,30)(19,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237740, #72237765, #72237766, …
maximal normal subgroup of	#72237740, #72237765, #72237766, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308676);