#458573

order	512
nilpotency class	3
isoclinism	isoclinic class of #458573
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16320
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

1146 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)(3,7,13,20)(4,8)(5,9)(6,10)(11,25,14,18)(12,19,28,31)(15,23)(16,22)(17,24)(21,29)(26,30,27,32);(1,3)(2,7,10,20)(4,14)(5,12)(6,13)(8,18,22,25)(9,19,24,31)(11,16)(15,27)(17,28)(21,30,23,32)(26,29);(1,4)(2,8)(3,11,13,14)(5,15)(6,16)(7,18,20,25)(9,23)(10,22)(12,27,28,26)(17,29)(19,30,31,32)(21,24);(1,5)(2,9)(3,12)(4,15)(6,17)(7,19)(8,21)(10,24)(11,26)(13,28)(14,27)(16,29)(18,30)(20,31)(22,23)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237746, #72237754, #72237762, …
maximal normal subgroup of	#72237746, #72237754, #72237762, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365769);