#210571

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

(2,4)(5,6,10,11,9,8,7,12)(13,14,18,22)(15,19,16,24)(17,26,27,21)(20,23,28,25);(1,2,3,4)(5,7,9,10)(6,11,8,12)(13,15,18,16)(14,19,22,24)(17,23,27,25)(20,26,28,21);(6,8)(11,12)(13,16,18,15)(14,20,22,28)(17,23,27,25)(19,26,24,21);(5,8,10,12,9,6,7,11)(15,16)(19,24)(20,28)(23,25)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=gh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dhi,d^c=dhi,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117767);