#6343220

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62759	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,e^b=ei,f^b=fh,d^c=dhi,e^c=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286982, #72319851, #72320117
maximal normal subgroup of	#72286982, #72319851, #72320117

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250416);