#396361

order	512
nilpotency class	3
isoclinism	isoclinic class of #396361
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14732
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1437 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)(3,13)(4,19,21,26)(5,11)(6,10)(7,12)(8,16)(9,22,25,20)(14,18)(15,29,31,23)(17,24)(27,32,28,30);(1,3,5,16,7,18,6,17)(2,8,10,13,12,24,11,14)(4,15,20,30,21,31,22,32)(9,23,19,27,25,29,26,28);(1,2)(3,14)(4,9)(5,10)(6,11)(7,12)(8,16)(13,18)(15,28)(17,24)(19,20)(21,25)(22,26)(23,30)(27,31)(29,32);(1,4,5,20,7,21,6,22)(2,9,10,19,12,25,11,26)(3,15,16,30,18,31,17,32)(8,23,13,27,24,29,14,28)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251786, #72251789, #72251798, …
maximal normal subgroup of	#72251786, #72251789, #72251798, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303557);