#387233

order	512
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

876 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=efg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=dei,f^c=fi,g^c=gi,e^d=ei,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72235145, #72235149, #72235150

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294429);