#209861

order	512
nilpotency class	3
isoclinism	isoclinic class of #18524
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#18524, #72279227, #72309465, …
maximal normal subgroup of	C₄×#18524, #72279227, #72309465, …
maximal quotient of	#72258279, #72258294, #72279611

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117057);