#6394041

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342836
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3520	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fgh,e^2=fi,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,g^c=gi,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268383, #72268384, #72268399, …
maximal normal subgroup of	#72268383, #72268384, #72268399, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6301237);