#497361

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#37674	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×D₄² ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1597 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 20 points:

(2,4)(5,6)(7,10)(8,12,16,11)(9,13)(14,18)(15,17,20,19);(1,2)(3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,17)(13,18)(16,20);(6,11)(7,14)(8,16)(10,17)(12,13)(18,19);(2,4)

Polycyclic presentation:

2^9|a^2=g,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234531, #72234532, #72234535, …
maximal normal subgroup of	#72234531, #72234532, #72234535, …
maximal quotient of	#72234578, #72317844, #72318046

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404557);