#494389

order	512
nilpotency class	4
isoclinism	isoclinic class of #33568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#33570, #72282140, #72282207, …
maximal normal subgroup of	C₄×#33570, #72282140, #72282207, …
maximal quotient of	#72282582

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401585);