#368513

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1425 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)(5,6,10,16,12,9,11,15)(7,17,19,18,20,14,8,13);(1,3,4,2)(5,7,12,20)(6,13,9,18)(8,10,19,11)(14,16,17,15);(5,8,11,7,12,19,10,20)(6,14,15,13,9,17,16,18);(5,9)(6,12)(7,18)(8,17)(10,15)(11,16)(13,20)(14,19)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#24472, #72312221, #72312233
maximal normal subgroup of	C₄×#24472, #72312221, #72312233

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275709);