#213847

order	512
nilpotency class	3
isoclinism	isoclinic class of #213828
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#2403	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,g^c=gi,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310955, #72310967, #72311008, …
maximal normal subgroup of	#72310955, #72310967, #72311008, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121043);