#485427

order	512
nilpotency class	4
isoclinism	isoclinic class of #485337
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1081 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)(3,12)(4,7)(5,9)(6,11)(8,14)(10,17)(13,25)(15,27)(16,22)(18,20)(19,21)(23,32)(24,29)(26,30)(28,31);(2,7)(4,17)(8,20)(9,10)(11,21)(12,27)(15,31)(18,32)(19,30)(22,28)(23,24)(25,26);(1,3,13,29)(2,8,9,24)(4,18,30,31)(5,16,6,14)(7,20,10,23)(11,12,25,22)(15,17,32,19)(21,27,26,28);(2,9)(3,14)(7,10)(8,22)(11,25)(12,24)(15,18)(16,29)(20,28)(21,26)(23,27)(31,32)

Polycyclic presentation:

2^9|c^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283632, #72338739
maximal normal subgroup of	#72283632, #72338739
maximal quotient of	#72284362

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392623);