#201388

order	512
nilpotency class	3
isoclinism	isoclinic class of #201384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1467 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18)(19,22)(20,27,28,23)(21,24)(25,29,31,32)(26,30);(1,3,5,11,6,12,4,13)(2,7,8,10,9,16,14,15)(17,19,21,26)(18,22,24,30)(20,25,28,31)(23,29,27,32);(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,31)(30,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240295, #72240297, #72240303, …
maximal normal subgroup of	#72240295, #72240297, #72240303, …
maximal quotient of	#72240338

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108584);