#365455

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

1519 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)(5,6,12,14)(7,15,13,19)(8,10,20,11)(9,18,16,17);(1,3,4,2)(5,7,12,13)(6,9,14,16)(8,19,20,15)(10,18,11,17);(5,8,10,14,12,20,11,6)(7,16,18,15,13,9,17,19);(5,9)(6,13)(7,14)(8,17)(10,19)(11,15)(12,16)(18,20)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#24472, #72312221, #72312300
maximal normal subgroup of	C₄×#24472, #72312221, #72312300

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272651);