#480405

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,8,5,6,3,7)(9,10)(11,21,24,36)(12,17)(13,18)(14,28,30,19)(15,20)(16,23)(22,37,26,40)(25,35,27,33)(29,32)(31,38)(34,39);(9,11)(10,16)(12,22)(13,26)(14,29)(15,24)(17,31)(18,34)(19,36)(20,32)(21,28)(23,30)(25,39)(27,38)(33,40)(35,37);(2,6)(7,8)(9,12,15,13)(10,17,20,18)(11,22,24,26)(14,25,30,27)(16,31,32,34)(19,33,28,35)(21,37,36,40)(23,38,29,39);(1,3,5,4)(2,7,6,8)(9,13,15,12)(10,18,20,17)(11,22,24,26)(14,27,30,25)(16,31,32,34)(19,35,28,33)(21,37,36,40)(23,38,29,39)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,e^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292957, #72292960, #72292963, …
maximal normal subgroup of	#72292957, #72292960, #72292963, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387601);