#213867

order	512
nilpotency class	3
isoclinism	isoclinic class of #213852
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1130 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)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(6,9)(14,15)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(20,28)(23,27)(25,32)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283499, #72283568, #72283815, …
maximal normal subgroup of	#72283499, #72283568, #72283815, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121063);